. 
< 


\mtm\i 


No 

Division 

Range 

Shelf. 

Received... 


PRESENTED  TO  THE 


If         I        li 


mm  of  tie  University  of  California 


UNIVERSITY   EDITION.— REVIStl)   AND   ENLARGED. 


TREATISE 


ASTRONOMY. 


DESCRIPTIVE.  THEORETICAL  AND  PHYSICAL, 


DESIGNED    FO* 


SCHOOLS,  ACADEMIES,  AND  PRIVATE  STUDENTS. 


BY  H.  N.  ROBINSON.  A.  M.. 

ff«MtM£2LY  PROFESSOR  OF  MATHEMATICS  IN  THE  UNITED  STATES  NAVY  ;    AUTHOR  Ol 

A  TREATISE  ON  ARITHMETIC,  iLGEBRA,  GEOMETRY,  TRIGONOMBTRT, 

SURVEYING.  CALCULUS.  NATURAL  PHILOSOPHY.  <&C.  <&C. 


NEW  YORK: 
IVISON,  PHINNEY,  BLAKEMAN  &  CO. 

CHICAGO:  S.  C.  GRIGGS  &  00. 
1866. 


Entered,  according  to  act  of  Congress  in  the  year  1849, 
BY  HORATIO  N   ROBINSON, 

In  the  Clerk's  Offier  or  the  District  Court  of  the  United  States,  for  the  District 
of  Ohio. 


Entered  according  to  act  of  Congress,  in  the  year  1857, 

BY  H.  N.  ROBINSON, 

U  the  Clerk's  Office  -If  the  District  Court  of  the  United  States  for  the  Northern 
District  of  Mew  York. 


PREFACE. 


To  give  at  once  a  clear  explanation  of  the  design  and  in- 
tended  character  of  this  work,  it  is  important  to  state  that  its 
author,  in  early  life,  imbibed  quite  a  passion  for  astronomy, 
and,  of  course,  he  naturally  sought  the  aid  of  books ;  but,  in 
this  field  of  research,  he  was  really  astonished  to  find  how 
little  substantial  aid  he  could  procure  from  that  source,  and 
not  even  to  this  day  have  his  desires  been  gratified. 

Then,  as  now,  books  of  great  worth  and  high  merit  were  to 
oe  found,  but  they  did  not  meet  the  wants  of  a  learner ;  the 
substantially  good  were  too  voluminous  and  mathematically 
abstruse  to  be  much  used  by  the  humble  pupil,  and  the  less 
mathematical  were  too  superficial  and  trifling  to  give  satis- 
faction to  the  real  aspirant  after  astronomical  knowledge. 

Of  the  less  mathematical  and  more  elaborate  works  on  as- 
tronomy there  are  two  classes — the  pure  and  valuable,  like 
the  writings  of  Biot  and  Herschel ;  but,  excellent  as  these 
are,  they  are  not  adapted  to  the  purposes  of  instruction ;  and 
every  effort  to  make  class  books  of  them  has  substantially 
failed.  From  the  other  class,  which  consists  of  essays  and 
popular  lectures,  little  substantial  knowledge  can  be  gathered, 
for  they  do  not  teach  astronomy ;  as  a  general  thing,  they  only 
glorify  it;  they  may  excite  our  wonder  concerning  the  im- 
mensity or  grandeur  of  the  heavens,  but  they  give  us  no  ad- 
ditional power  to  investigate  the  science. 

Another  class  of  more  brief  and  valuable  productions  were, 
and  are  always  to  be  found,  in  which  most  of  the  important 
facts  are  recorded ;  such  as  the  distances,  magnitudes,  and  mo- 
tions of  the  heavenly  bodies;  but  how  these  facts  became 
known  is  rarely  explained :  this  is  what  the  true  searcher  after 
science  will  always  demand,  and  this  book  is  designed  ex- 
pressly to  meet  that  demand. 

In  the  first  part  of  the  book  we  suppose  the  reader  entirely 
unacquainted  with  the  subject ;  but  we  suppose  him  compe- 
tent to  the  task — to  be,  at  least,  sixteen  years  of  age — to  have 
a  good  knowledge  of  proportion,  some  knowledge  of  algebra, 
geometry,  and  trigonometry — and  then,  and  not  until  then, 
can  the  study  be  pursued  with  any  degree  of  success  worth 
mentioning.  Such  a  person,  and  with  such  acquirements  as 

iii 


iv  PREFACE. 

PRKJMCK.  we  have  here  designated,  we  believe,  can  take  this  book  and 
learn  astronomy  in  comparatively  a  short  time;  for  the  chief 
design  of  the  work  is,  to  teach  whoever  desires  to  learn :  and 
it  matters  not  where  the  learner  may  be,  in  a  college, 
academy,  school,  or  a  solitary  student  at  home,  and  alone  in 
the  pursuit. 

The  book  is  designed  for  two  classes  of  students — the  well 
prepared  in  the  mathematics,  and  the  less  prepared ;  the  for- 
mer are  expected  to  read  the  text  notes,  the  latter  should 
omit  them.  With  the  text  notes,  we  conceive  it,  or  rather 
designed  it  to  be,  a  very  suitable  book  to  give  sound  elemen- 
tary instruction  in  astronomy ;  but  we  do  not  offer  the  work 
as  complete  on  practical  astronomy ;  for  whoever  becomes  a 
practical  astronomer  will,  of  course,  seek  the  aid  of  complete 
and  elaborate  sets  of  tables,  such  as  would  be  improper  to 
insert  in  a  school  book. 

We  have  inserted  tables  only  for  the  purpose  of  carrying 
out  a  sound  theoretical  plan  of  instruction,  and,  therefore,  we 
have  given  as  few  as  possible,  and  those  few  in  a  very  con- 
tracted form.  The  epochs  for  the  sun  and  moon  may  be  ex- 
tended forward  or  backward,  to  any  extent,  by  any  one  who 
understands  the  theory. 

The  chapters  on  comets,  variable  stars,  &c.,  are  compila- 
tions, and  are  printed  in  smaller  type ;  and  the  works  to 
which  we  are  most  indebted,  are  Hcrschel's  Astronomy  and 
the  Cambridge  Astronomy,  originally  the  work  of  M.  Biot. 

Other  parts  of  the  work,  we  believe,  will  be  admitted  as 
mainly  original,  by  all  who  take  pains  to  examine  it. 

The  chief  merits  claimed  for  this  book  are,  brevity,  clear- 
ness of  illustration,  anticipating  the  difficulties  of  the  pupil, 
and  removing  them,  and  bringing  out  all  the  essential  points 
of  the  science. 

Some  originality  is  claimed,  also,  in  several  of  our  illustra- 
tions, particularly  that  of  showing  the  rationale  of  tides  rising 
on  the  opposite  sides  of  the  earth  from  the  moon ;  and  in  tho 
general  treatment  of  eclipses ;  but  it  is  for  others  to  deter- 
mine how  much  merit  should  be  awarded  for  such  originali- 
ties; we  have,  however,  used  greater  conciseness  and  per- 
spicuity in  general  computations  than  is  to  be  found  in  most 
of  the  books  on  this  subject ;  and  this  last  remark  will  apply 
to  the  whole  work. 


PREFACE    TO    THE    REVISED    EDITION. 


THE  author  and  publisher  of  this  volume  have  tlio  satisfaction  of    pRKFACE 

knowing,  that  all  teachers,  who  are  really  qualified  to  teach  Astron-~ 

omy,  and  who  have  examined  this  hook,  hold  it  in  the  highest  estim- 
ation ;  and  they  only  regret  that  so  few  pupils  are  prepared  to  profit 
by  it. 

This  being  more  complimentary  than  discouraging,  we  resolved  to 
make  the  book  as  useful  as  possible,  in  the  hope  that  it  will  aid  in 
imparting  essential  knowledge  of  Astronomy  to  many  of  the  youths 
of  the  United  States. 

With  this  object  in  view  we  have  increased  the  present  edition  by 
fifty  pages  of  very  practical  and  important  matter. 

In  the  former  editions  we  were  careful  not  to  give  more  than  could 
be  received,  and  the  subject  of  solar  eclipses  was  not  exhausted. 

In  the  Nautical  Almanacs,  a  rude  map  generally  represents  the  parts 
of  the  earth  over  which  tho  solar  eclipses  will  be  visible,  and  curved 
lines  and  loops  mark  tlu  places  where  the  eclipse  will  commence  at 
sunrise,  and  end  with  sunset,  <tc.,  <fec.,and  the  student  may  naturally 
ask,  how  these  lines  arc  determined  ? 

In  this  volume  we  have  attempted  to  give  the  key  to  the  whole 
solution,  and  we  shall  feel  disappointed  if  our  efforts  are  pronounced 
abortive. 

Professor  J.  R.  McFarland,  of  Oxford,  Ohio,  a  zealous  lover  of 
science,  requested  us  to  add  rules  for  computing  the  times  of  rising 
and  setting  of  the  moon  and  stars  ;  and  also  to  explain  the  Argu- 
ments for  the  twenty  small  equations  of  the  moon's  longitude.  We 
cheerfully  complied  with  his  requests.hopingthatthose  additions  will 
l«?  as  welcome  to  others, as  to  him. 

EI.DRTDGE,  N.  Y.,  July,  1857.  r 


CONTENTS. 


SECTION  L 

r.f. 

liCTEODucriON. — Definition  of  terms,  &c., 11 

CHAPTER  I. 

Preliminary  Observations, 16 

A  fixed  point  in  the  heavens — the  pole  and  polar  star, 17 

Index  to  the  length  of  one  year, 21 

Fixed  stars — why  so  called, 22 

CHAPTER    II. 

Appearances  i  n  the  heavens 23 

Importiuii  instruments  for  an  Observatory, 24 

Standard  measure  for  time, 26 

An  astronomical  clock, 27 

Movable  and  wandering  bodies, 29 

To  find  the  right  ascension  of  the  sun,  moon,  and  planets, 31 

CHAPTER   HI. 

Refraction — position  of  the  equinox,  &c., 31 

Altitude  and  azimuth  instrument 33 

Astronomical  refraction  —  its  effect,  &c., 35 

The  declination  of  a  star — how  found, 39 

Observations  to  find  the  equinox, 43 

Length  of  the  year — how  observed, 46 

CHAPTER  IV. 

Geography  of  the  heavens, 48 

Method  of  tracing  the  stars, 51 

What  constitutes  a  definite  description, 53 

How  to  find  the  right  ascension  of  any  star, 54 

The  southern  cross  and  Magellan  clouds, 58 


SECTION  II. 
DESCRIPTIVE   ASTRONOMY. 

CHAPTER  I. 
First  consideration  of  the  distances  to  the  heavenly  bodies — also 

and  figure  of  the  earth, 59 

How  to  find  the  diameter  of  the  earth... 63 


CONTENTS. 


Dip  of  the  horizon,  .........................................  64 

The  exact  dimensions  of  the  earth,  ...............  ,  ...........  65    Coin-tin*. 

Gravity  of  the  earth  diminished  on  its  surface  by  its  rotation,...  68 

A  degree  between  two  meridians  —  the  law  of  decrease,  .........  71 

CHAPTER  II. 

Parallax,  general  and  horizontal,  .........................  ....  72 

Relation  between  parallax  and  distance,  ..............  .  .......  .  73 

Lunar  parallax  —  how  found,  .....  ......................  .....  75 

Variable  distance  to  the  moon,  .....  ................  ....  ......  77 

Apogee  and  perigee,  ........................................  77 

Mean  parallax  —  and  parallax  at  mean  distance,  ...............  78 

Mean  distance  to  the  moon,  ..................  .............  ..  79 

Connection  between  semidiarneter  and  the  horizontal  parallax  of 

any  celestial  body,  ..........  .  .....  .  ......................  80 

The  earth  a  moon  to  the  moon,  ..............................  81 

CHAPTER  III.  ,« 

The  earth's  orbit  eccentric,  &c.,  .......  .  .....................     82 

Methods  of  measuring  apparent  diameters,...  .  ................     83 

Eccentricity  of  the  earth's  orbit  —  how  found,  ................     86 

Variations  in  solar  motion,  ...........  .  ......................     86 

Eccentricity  of  orbit  and  greatest  equation  of  center  connected,.  .     95 

CHAPTER  IV. 
Causes  of  the  change  of  seasons,  ................  .............     97 

Temperature  of  the  earth,  ..................................     99 

Times  of  extreme  temperature,  .........  .....................  100 

&)i     *«.<•<•...«..*..  t  ......  , 

CHAPTER  V. 

Equation  of  time  ...........................................  100 

Mean  and  apparent  noon,  ...................................  101 

What  is  meant  by  sun  slow  and  sun  fast,  .............  .  ........  104 

Use  of  the  equation  of  time,  ................................  106 

CHAPTER   VI. 
Apparent  motions  of  the  planets........  .....  ................  107 

The  morning  and  evening  star,  ..............................  108 

Motion  of  Venus  in  respect  to  the  fixed  stars,  .................   Ill 

Retrograde  motion  of  planets  accounted  for  ...................  112 

CHAPTER  VII. 

First  approximations  of  the  relative  distances  of  the  planets  from 
the  sun,  .................................................  114 

What  to  understand  by  stationary,  ...........................  116 

Method  of  approximating  to  the  orbits  of  planets,.  .  .  ...........  118 


VU!  CONTENTS. 

CHAPTER    VIII. 

Paf. 

CONTENTS    Methods  of  observing  the  periodical  revolutions  of  the  planets,..  121 

Diurnal  motion  of  the  planets, 124 

Times  of  revolution  and  distances  compared 127 

Kepler's  Laws, 128 

CHAPTER   IX. 

Transits  of  Venus  and  Mercury, 129 

Periods  of  the  transits  of  Venus 130,  131,  130 

Deductions  from  a  transit  made  plain, 134 

CHAPTER  X. 

The  horizontal  parallaxes  of  the  planets  computed, 137 

Real  distance  between  the  earth  and  sun  determined, 138 

How  to  find  the  magnitudes  of  the  planets, 140 

CHAPTER   XI. 

A  general  description  of  the  planets, 141 

Professor  Bode's  law  of  planetary  distances, 144 

A  bold  hypothesis, 145 

Progressive  nature  of  light  —  how  determined, 149 

CHAPTER   XII. 
Comets, 1 54 

Inclinations  of  their  orbits, 157 

Fears  anciently  entertained  concerning  comets, 160 

CHAPTER  XIII. 
On  the  peculiarities  of  the  fixed  stars, 160 


SECTION  m. 

PHYSICAL   ASTRONOMY. 

CHAPTER   I. 

General  laws  of  motion — theory  of  gravity, 167 

Attraction  of  a  sphere — of  a  spherical  shell,  &c., 171 

A  general  expression  for  the  mutual  attraction  of  two  bodies,..,.  172 

CHAPTER  II. 

Demonstration  of  Kepler's  Laws, 173 

A  common  error, 177 

How  a  planet  finds  its  own  orbit 178 

Kepler's  third  law  rigorously  true  in  circles  and  ellipses, 181 


CONTENTS  U 

CHAPTER   III. 

P*f«.  CONTENT*. 

Masses  of  the  planets,  &c., 184 

The  diameter  of  the  earth  accurately  determined  from  equations 

in  physical  astronomy,  (  Art.  171 ), 190 

The  mass  of  the  moon  determined — densities  of  bodies, 193 

CHAPTER   IV. 

Lunar  Perturbations, 195 

Cause  of  nutation, 197 

Mean  radial  force, 201 

Acceleration  of  the  moon's  mean  motion, 307 

A  summary  statement  of  the  cause, 208 

The  true  mean  value  of  the  radial  force, 209 

A  summary  statement  of  the  lunar  irregularities, 210 

CHAPTER   V. 

The  tides, 211 

A  summary  illustration  of  the  physical  cause  of  tides 212 

Mass  of  the  moon  computed  from  the  tides, 214 

CHAPTER   VI. 

Planetary  perturbations, 216 

Action  and  reaction  equal  and  contrary 216 

The  effects  of  commensurate  revolutions  of  the  planets, 218 

The  great  inequalities  of  Jupiter  and  Saturn, 219 

These  inequalities  explained, 219,  220,  221,  222 

The  physical  effects  on  Uranus  that  led  to  the  discovery  of  Neptune,  223 

CHAPTER   VII. 

Aberration — nutation,  and  precession  of  the  equinoxes, 223 

The  velocity  of  light  computed  from  the  effect  of  aberration,... .  225 

Cause  of  nutation  explained 228 

The  physical  cause  of  the  precession  of  the  equinoxes, 232 

Proper  motion  of  the  stars — how  found, 234 

The  latitude  of  the  sun  explained, 236 


SECTION  IV. 

PRACTICAL   ASTRONOMY. 
Preparatory  remarks  and  trigonometrical  formula?, 239 

CHAPTER  I. 

Problems  in  relation  to  the  sphere, 242 

To  find  the  time,  from  the  latitude  of  the  place — altitude,  and 
declination  of  the  BUB 249 


x  CONTENTS. 

COMT 

An  artificial  horizon, 253 

Absolute  and  local  time, 254 

Lunar  observations, 256 

Proportional  logarithms, 257 

CHAPTER  II. 

Explanation  of  the  tables, 260 

To  compute  the  sun's  longitude, 263 

To  find  the  equation  of  time  to  great  exactness, 264 

To  compute  the  time  of  new  and  full  moons,, 265 

Eclipses — when  they  occur,  &c., 267 

Limits  of  eclipses,. 269 

Periods  of  eclipses, 271,  273 

Elements  of  lunar  eclipses, 274 

Semidiameter  of  the  earth's  shadow, 275 

CHAPTER   III. 

Preparation  for  the  computation  of  eclip  es, 277 

Directions  for  computing    he  moon's  longitude,  latitude,  &c.,. ..  278 

To  construct  a  lunar  eclipse, 285 

To  make  exact  computations  in  respect  to  lunar  eclipses, 2b7 

CHAPTER   IV. 

Solar  eclipses — general  and  local, 289 

Elements  for  the  computation  of  a  solar  eclipse, 290 

To  construct  a  general  eclipse 291 

How  to  determine  the  duration  of  a  solar  eclipse  on  the  earth,. ..  292 

To  find  where  the  sun  will  be  centrally  eclipsed  at  noon, 296 

Results  taken  from  the  projection, 297 

Results  from  trigonometrical  computations, 298 

CHAPTER   V. 
Local  .eclipses,  &c., 301 

How  to  construct  a  local  solar  eclipse, 301 

How  to  find  the  time  of  greatest  obscuration, 305 

To  find  the  time  of  the  beginning,  end,  &c.,  of  a  local  eclipse  by 
the  application  of  analytical  geometry, 306 

U^For  contents  of  Appendix,  see  page  358, 


ASTRONOMY. 


INTRODUCTION. 

ASTRO  KOMY  is  the  science  which  treats  of  the  heavenly    A§tw>no»y 
todies,  describes  their  appearances,  determines  their  magni-  defined- 
tudes,  and  discovers  the  laws  which  govern  their  motions. 

When  we  merely  state  facts  and  describe  appearances  a"s    The  dm. 
they  exist  in  the  heavens,  we  call  it  Descriptive  Astronomy.  8I< 
When  we  compute  magnitudes,  determine  distances,  record 
observations,  and  make  any  computations  whatever,  we  call 
it  Practical  Astronomy. 

The  investigation  of  the  laws  which  govern  the  celestial 
motions,  and  the  explanation  of  the  causes  which  bring  about 
the  known  results,  is  called  Physical  Astronomy. 

When  the  mariner  makes  use  of  the  index  of  the  heavens,     Nautical 
to  determine  his  position  on  the  earth,  such  observations,  and  M 
their  corresponding  computations,  are  called  Nautical  Astro- 
nomy. 

By  nautical  astronomy  we    determine  positions   on   the    Geography 
earth,  and  subsequently,  the  magnitude  of  the  earth ;  and 
thus,  we  perceive,  that  Geography  and  Astronomy  must  be 
linked  together ;  and  no  one  can  fully  understand  the  former 
science,  without  the  aid  of  the  latter. 

Astronomy  is  the  most  ancient  of  all  the  sciences,  for,  in  The  ami- 
the  earliest  age,  the  people  could  not  have  avoided  observing 
the  successive  returns  of  day  and  night,  and  summer  and 
winter.  They  could  not  fail  to  perceive  that  short  days  cor- 
respon  ded  to  winter,  and  long  days  to  summer ;  and  it  was 
thus,  probably,  that  the  attentions  of  men  were  first  drawn 
to  the  study  of  astronomy. 

,  11 


1*  ASTRONOMY. 

iKxaoDtrc,       in  tins  work,  we  shall  not  take  facts  unless  they  are  within 
Facu  aion«  the  sphere  of  our  own  observations.     We  shall  not  perempto- 
aot  science.  ^  gtate  ^^  ^Q  earth,  is  7912  miles  in  diameter;  that  the 
moon  is  about  240,000  miles  from  the  earth,  and  the  sun 
95,000,000  of  miles ;  for  such  facts,  alone,  and  of  themselves,  do 
not  constitute  knowledge,  though  often  mistaken  for  knowledge. 
We  shall  direct  the  mind  of  the  reader,  step  by  step,  through 
the  observations  and  through  the  investigations,  so  that  he 
can  decide  for  himself  that  the  earth  must  be  of  such  a  mag- 
nitude, and  is  thus  far  from  the  other  heavenly  bodies ;  and 
that  will  be  knowledge  of  the  most  essential  kind. 
Th«  fowl.      Al}  astronomical  knowledge  has  its  foundation  in  observa- 
Mtamomfcai  **on '  an<*  ^e  ^*st  object  of  this  book  shall  be  to  point  out 
knowledge,   what  observations  must  be  taken,  and  what  deductions  must 
be  made  therefrom ;  but  the  great  book  which  the  pupil  must 
study,  if  he  would  meet  with  success,  is  the  one  which  spreads 
out  its  pages  on  the  blue  arch  above ;  and  he  must  place  but 
secondary  dependence  on  any  book  that  is  merely  the  work 
of  human  art. 

As  we  disapprove  of  the  practice  of  throwing  to  the  reader 
astounding  astronomical  facts,  whether  he  can  digest  them  or 
not,  and  as  we  are  to  take  the  inductive  method,  and  to  lead 
the  student  by  the  hand,  we  must  commence  on  the  supposi- 
tion that  the  reader  is  entirely  unacquainted  even  with  the 
common  astronomical  facts,  and  now  for  the  first  time  seriously 
brings  his  mind  to  the  study  of  the  subject ;  but  we  shall 
suppose  some  maturity  of  mind,  and  some  preparation,  by  the 
acquisition  of  at  least  respectable  mathematical  knowledge, 
c-onren-  Every  science  has  its  technicalities  and  conventional  terms ; 
and  definu  an<^  astronomy  is  by  no  means  an  exception  to  the  general 
rule  ;  and  as  it  will  prepare  the  way  for  a  clearer  understand- 
ing of  our  subject,  we  now  give  a  short  list  of  some  of  the 
technical  terms,  which  must  be  used  in  our  composition. 

Horizon. — Every  person,  wherever  he  maybe,  conceives 
himself  to  be  in  the  center  of  a  circle;  and  the  circumference 
of  that  circle  is  where  the  earth  and  sky  apparently  meet. 
That  circle  is  called  the  horizon. 


INTRODUCTION.  13 


Altitude.  —  The  perpendicular  bight  from  the  horizon, 
measured  by  degrees  of  a  circle. 

Meridian.  —  An  imaginary  line,  north  and  south  from  any 
point  or  place,  whether  it  is  conceived  to  run  along  the  earth 
or  through  the  heavens.  If  the  meridian  is  conceived  to 
divide  both  the  earth  and  the  heavens,  it  is  then  considered 
as  a  plane,  and  is  spoken  of  as  the  plane  of  the  meridian. 

Poles.  —  The  points  where  all  meridians  come  together  : 
poles  of  the  earth  —  the  extremities  of  the  earth's  axis. 

Zenith.  —  The  zenith  of  any  place,  is  the  point  directly    Poiei    «f 
overhead  ;  and  the  Nadir  is  directly  opposite  to  the  zenith,  or  tie  homOB* 
under  our  feet.     The  zenith  and  nadir  are  the  poles  to  the 
horizon. 

Verticals.  —  All  lines  passing  from  the  zenith,  perpendicu-  (  Pn»«  vw 
lar  to  the  horizon,  are  called  ^7erticals,  or   Vertical  Circles.  tlcal> 
The  one  passing  at  right  angles  to  the  meridian,  and  striking 
the  horizon  at  the  east  and  west  points,  is  called  the  Prime 
Vertical. 

Azimuth.  —  The  angular  position  of  a  body  from  the  meri- 
dian, measured  on  the  circle  of  the  horizon,  is  called  its  Azi- 
muth. 

The  angular  position,  measured  from  its  prime  vertical,  is 
called  its  Amplitude. 

The  sum  of  the  azimulh  and  amplitude  must  always  make 
90  degrees. 

Equator.  —  The  Earth's  Equator  is  a  great  circle,  east  and 
west,  and  equidistant  from  the  poles,  dividing  the  earth  into 
two  hemispheres,  a  northern,  and  a  southern. 

The  Celestial  Equator  is  the  plane  of  the  earth's  equator 
conceived  to  extend  into  the  heavens.  equator. 

When  the  sun,  or  any  other  heavenly  body,  meets  the      Equ 
celestial  equator,  it  is  said  to  be  in  the  Equinox,  and  the  tw* 
equatorial  line  in  the  heavens  is  called  the  Equinoctial. 

Latitude.  —  The  latitude  of  any  place  on  the  earth,  is 
its  distance  from  the  equator,  measured  in  degrees  on  the 
meridian,  either  north  or  south. 

If  the  measure  is  toward  the  north,  it  is  north  latitude;  if 
toward  the  couth,  south  latitude. 


»4  ASTRONOMY. 

**r»oprc.  The  distance  from  the  equator  to  the  poles  is  90  degrees — 
one- fourth  of  a  circle;  and  we  shall  know  the  circumference 
of  the  whole  earth,  whenever  we  can  find  the  absolute  length  of 
one  degree  on  its  surface. 

Co-Latitude.  —  Co-latitude  is  the  distance,  in  degrees,  of 
any  place  from  the  nearest  pole. 

The  latitude  and  co-latitude  (  complement  of  the  latitude) 
must,  of  course,  always  make  90  degrees. 

Parallels      Parallels  of  latitude  are  small  circles  on  the  surface  of  the 
of  latitude    earth,  parallel  to  the  equator. 

Every  point,  in  such  a  circle,  has  the  same  latitude. 
Longitude.  —  The  longitude  of  a  place,  on  the  surface  of 
the  earth,  is  the  inclination  of  its  meridian  to  some  other 
meridian  which  may  be  chosen   to   reckon   from.     English 
astronomers  and  geographers  take  the  meridian  which  runs 
through  Greenwich  Observatory,  as  the  zero  meridian. 
TL«  fir«t      Other  nations  generally  take  the  meridian  of  their  princi- 
••ridiaii  ar-  paj  observatories,  or  that  of  the  capital  of  their  country,  ag 
the  first  meridian;  but  this  is  national  vanity,  and  creates 
only  trouble  and  confusion :  it  is  important  that  the  wholt 
world  should  agree  on  some  one  meridian,  from  which  to  reckon 
longitude ;  but  as  nature  has  designated  no  particular  one,  it 
is  not  wonderful  that  different  nations  have  chosen  different 
lines. 

w«  adopt      jn  this  work,  we  shall  adopt  the  meridian  of  Greenwich  a« 
•f     Green- tne  zero  ^ne  °f  longitude,  because  most  of  the  globes  and 
wich ;    and  maps,  and  all  the  important  astronomical  tables,  are  adapted 
wh7 1          to  that  meridian,  and  we  see  nothing  to  be  gained  by  chang- 
ing them. 

Declination.  —  Declination  refers  only  to  the  celestial  equa- 
tor, and  is  a  leaning  or  declining,  north  or  south  of  that  line, 
and  is  similar  to  latitude  on  the  earth. 

Solstitial  Points.  —  The  points,  in  the  heavens,  north  and 
south,  where  the  sun  has  its  greatest  declination. 

The  northern  point  we  call  the  Summer  Solstice,  and  the 
southern  point  the  Winter  Solstice;  the  first  is  in  longitude 
90°,  the  other  in  longitude  270°. 

As  latitude  is  reckoned  north  and  south,  so  longitude  is 


INTRODUCTION.  |6 

reckoned  east  and  west ;  but  it  would  add  greatly  to  syste-   INTRUDUO. 
matic  regularity,  and  tend  much  to  avoid  confusion  and  am-       improv«. 
biguity  in  computations,  were  this  mode  of  expression  aban-  ment     rag. 
doned,  and  longitude  invariably  reckoned  westward,  from  0  to  se*ted* 
360  degrees. 

Latitude  and  longitude,  on  the  earth,  doeer  not  corre-      Latitude, 
spond  to  latitude  and  longitude  in  the  heavens.     Latitude,  on  ^'^ tTt  a»- 
the  earth,  corresponds  with  declination  in  the  heavens ;  and  cension. 
longitude,  on  the  earth,  has  a  striking  analogy  to  right  ascen- 
sion  in  the  heavens,  though  not  an  exact  correspondence. 
We  shall  more  particularly  explain  latitude,  longitude,  and 
right  ascension  in  the  heavens,  as  we  advance  in  this  work ; 
for  it  is  only  when  we  are  forced  to  use  these  terms,  that  the 
nature  and  spirit  of  their  import  can  be  really  understood. 

There  are  other  technicalities,  and  terms  of  frequent  use,    other  term* 
in  astronomy,  such  as  Conjunction,  Opposition,  Retrograde,  "°*  explaiap 
Direct,  Apogee,  Perigee,  &c.,  &c.,  all  of  which,  for  the  sake 
of  simplicity,  had  better  not  be  explained  until  they  fall 
into  use ;  and,  once  for  all,  let  us  impress  this  fact  on  the 
Hinds  of  our  readers,  that  we  shall  put  far  more  stress  on  the 
substance  ind  spirit  of  a  thing,  than  on  its  name. 


6  ASTRONOMY. 


SECTION  I. 


CHAPTER   I. 

PRELIMINARY   OBSERVATIONS. 

CHAT,  i.        To  commence  the  study  of  astronomy,  we  must  observe 
and  call  to  mind  the  real  appearances  of  the  heavens. 

Take  such  a  station,  any  clear  night,  as  will  command  an 
extensive  view  of  that  apparent,  concave  hemisphere  above 
us,  which  we  call  the  sky,  and  fix  well  in  the  mind  the  direc- 
tions of  north,  south,  east,  and  west. 

The  appa-      At  first,  let  us  suppose  our  observer  to  be  somewhere  in 
•f  the  Stan.  tn®  United  States,  or  somewhere  in  the  northern  hemisphere, 
about  40  degrees  from  the  equator. 

As  yet,  this  imaginary  person  is  not  an  astronomer,  and 
neither  has,  nor  knows  how  to  use,  any  astronomical  instru- 
ment ;  but  we  would  have  him  mark  with  attention  the  po- 
sitions of  the  heavenly  bodies. 

( 1. )  Soon  he  will  perceive  a  variation  in  the  position  of 
the  stars :  those  at  the  east  of  him  will  apparently  rise ;  those 
at  the  west  will  appear  to  sink  lower,  or  fall  below  the  hori- 
zon ;  those  at  the  south,  and  near  his  zenith,  will  apparently 
move  westward;  and  those  at  the  north  of  him,  which  he  may 
see  about  half  way  between  the  horizon  and  zenith,  will  appear 
stationary. 

Apparent  Let  such  observations  be  continued  during  all  the  hours 
°^  ^  nign*>  an^  for  several  nights,  and  the  observer  cannot 
fail  to  be  convinced  that  not  only  all  the  stars,  but  the  sun, 
moon,  and  planets,  appear  to  perform  revolutions,  in  about 
twenty-four  hours,  round  &  fixed  point;  and  that  fixed  point, 
as  appears  to  us  (in  the  middle  and  northern  part  of  the 
United  States  ),  is  about  midway  between  the  northern  hori- 
zon and  the  zenith. 

ft  gnoui  J  always  be  borne  in  mind,  that  the  sun,  moon,  and 
clro!"'  gtars,  have  an  apparent  diurnal  motion  round  a  fixed  point, 


PRELIMINARY    OBSERVATIONS.  i7 

and  all  those  stars  which  are  90  degrees  from  that  point,    CHAP.  i. 
apparently  describe   a  great  circle.     Those  stars   that    are 
nearer  to  the  fixed  point  than  90  degrees,  describe  smaller 
circles;  and  the  circles  are  smaller  and  smaller  as  the  objects 
are  nearer  and  nearer  the  fixed  points. 

(  2.  )  There  is  one  star  so  near  this  fixed  point,  that  the 
small  circle  it  describes,  in  about  24  hours,  is  not  apparent 
from  mere  inspection.  To  detect  the  apparent  motion  of 
this  star,  we  must  resort  to  nice  observations,  aided  by  ma- 
thematical instruments. 

This  fixed  point,  that  we  have  several  times  mentioned,  is    The  North 
the  North  Pole  of  the  heavens,  and  this  one  star  that  we  have  just  Star* 
mentioned,  is  commonly  called  the  North  Star,  or  the  Pole  Star. 

(3.)  This  star,  on  the  1st  of  January,  1820,  was  1°  39'    Po«Uo»of 
6"  from  the  pole,  and  on  1st  of  January,  1847,  its  distance  *he     Nort* 
from  the  pole  was    1°  30'    8";    and  it  will  gradually  and 
more  slowly  approach  within  about  half  a  degree  of  the  pole, 
and  afterward  it  will  as  gradually  recede  from  the  pole,  and 
finally  cease  to  be  the  polar  star. 

We  here,  and  must  generally,  speak  of  the  star,  or  the  stars,     The    poll 
as  in  motion ;  but  this  is  not  so.     The  fixed  stars  are  abso-  m  motl°u' 
iuiety  fixed ;  it  is  the  pole  itself  that  has  a  slow  motion  among 
the  stars,  but  the  cause  of  this  motion  cannot  now  be  ex- 
plained; it  is  one  of  the  most  abstruse  points  in  astronomy, 
and  we  only  mention  it  as  a  fact. 

As  the  North  Star  appears  stationary,  to  the  common  ob- 
server, it  has  always  been  taken  as  the  infallible  guide  to 
direction ;  and  every  sailor  of  the  ocean,  and  every  wanderer 
of  the  African  and  Arabian  deserts,  has  held  familiar  ac- 
quaintance with  it. 

(  4. )  If  our  observer  now  goes  more  to  the  southward,  and    changes  oi 
makes  the  same  observations  on  the  apparent  motions  of  the  aPPearanr« 
stars,  he  will  find  the  same  general  results ;  each  individual  ,0othwaid. 
star  will  describe  the  same  circle ;  but  the  pole,  the  fixed 
point,  will  be  lower  down,  and  nearer  the  northern  horizon ; 
and  it  will  be  lower  and  lower  in  proportion  to  the  distance 
the  observer  goes  to  the  south.     After  the  observer  has  gone 
sufficiently  far,  the  fixed  point,  the  pole,  will  no  longer  be  up 
2) 


18  ASTRONOMY. 

CHAP.  I.    in  the  heavens,  but  down  in  the  northern  horizon  ;  and  when 

Appear-  the  pole  does  appear  in  the  horizon,  the  observer  is  at  the 

aace     from  equator,  and  from  that  line  all  the  stars  at  or  near  the  equa- 

the  eqn*tor.  ^  appear  to  rise  up  directly  from  the  east,  and  go  down 

directly  to  the  west;  and  all  other  stars,  situated  out  of  the 

equator,  describe  their  small  circles  parallel  to  mis  perpendi- 

cular equatorial  circle. 

South    of      jf  ^e  Okserver  g0es  south  of  the  equator,  the  apparent 
north  pole  of  the  heavens  sinks  below  the  northern  horizon, 
and  the  south  pole  rises  up  into  the  heavens  at  the  south. 
Changes  in      /  5  \  jf  fae  observer   should   go   north,  from   the   first 

appearance  .  .  •        »  •,  -i  i       • 

on       going  station,  m  place  of  going  south,  the  north  pole  would  rise 
north  nearer  to  the  zenith  ;  and,  should  he  continue  to  go  north,  he 

would  finally  find  the  pole  in  his  zenith,  and  all  the  stars 
would  apparently  make  circles  round  the  zenith,  as  a  center, 
and  parallel  to  the  horizon  ;  and  the  horizon  itself  would  be  the 
celestial  equator. 

(  6.  )  When  the  north  pole  of  the  heavens  appears  at  the 
zenith,  the  observer  must  then  be  at  the  north  pole,  on  <he 
earth,  or  at  the  latitude  of  90  degrees. 


Appear.      ^  7  ^  ^ny  celestial  body,  which  is  north  of  the  equator,  is 

the      north  always  visible  from  the  north  pole  of  the  earth  ;  hence  the 

pole.  gun,  which  is  north  of  the  equator  from  the  20th  of  March  to 

the  23d  of  September,  must  be  constantly  visible  during  that 

period,  in  a  clear  sky. 

Just  as  the  sun  comes  north  of  the  equator,  its  diurnal 
progress,  or  rather,  the  progress  of  24  hours,  is  around  the 
horizon.  When  the  sun's  declination  is  10  degrees  north  of 
the  equator,  the  progress  of  24  hours  is  around  the  horizon 
at  the  altitude  of  10  degrees  ;  and  so  for  any  other  degree. 

From  the  north  pole,  all  directions,  on  the  surface  of  the 
earth,  are  south.     North  would  be  in  a  vertical  direction 
toward  the  zenith. 
HOW    to      we  have  observed  that  the  pole  of  the  heavens  rises  as  we 

find   the  cir-  i.,  »     «      «*       •  ±\.  4 

conference    go  north,  and  sinks  toward  the  horizon  as  we  go  south  ;  ana 


anddiameter  wnen  we  observe  that  the  pole  has  changed  its  position  one 
degree,  in  relation  to  the  horizon,  we  know  that  we  mui 
changed  place  one  degree  on  the  surface  of  the  earth. 


PRELIMINARY    OBSERVATIONS.  19 

(  8. )  Now  we  know  by  observation,  that  if  we  go  north  CHAP,  i 
about  69i  English  miles  on  the  earth,  the  north  pole  will  be 
one  degree  higher  above  the  horizon.  Therefore  69i  miles 
corresponds  to  one  degree,  on  the  earth ;  and  hence  the  whole 
circumference  of  the  earth  must  b«j  69i  multiplied  by  360 : 
for  there  are  360  degrees  to  every  circle.  This  gives  24,930 
miles  for  the  circumference  of  the  earth,  and  7,930  miles  for 
its  diameter,  which  is  not  far  from  the  truth. 

(  9. )  Here,  in  the  United  States,  or  anywhere  either  in     circumpo. 
Europe,  Asia,  or  America,  north  of  the  equator,  say  in  lati-  lar  stars- 
tude  40°,  the  north  pole  of  the  heavens  must  appear  at  an 
altitude  of  40°  above  the  horizon ;  and  as  all  the  stars  and 
heavenly  bodies  apparently  circulate  round  this  point  as  a 
center,  it  follows  that  all  those  stars  which  are  within  409 
of  the  pole  can  never  go  below  the  horizon,  but  circulate 
round  and  round  the  pole.     All  those  stars  which  never  go 
below  the  horizon,  are  called  drcumpolar  stars. 

At  the  north,  and  very  near  the  north  pole,  the  sun  is  a  The  snn  a 
drcumpolar  body  while  it  is  north  of  the  equator,  and  it  is  a  JJJJ™™1^^ 
circumpolar  body  as  seen  from  the  south  pole,  while  it  is  south  from  th« 
of  the  equator:  this  gives  six  months  dav  and  six  months  north of  lati' 

.    ,  J  tude  6C    de- 

night,  at  the  poles.  grees< 

(  10. )  North  of  latitude  66°,  and  when  the  sun's  declina- 
nation  is  more  than  23°  north  (  as  it  is  on  and  about  the  20th 
of  June  in  each  year  ),  then  the  sun  comes  at,  or  very  near,  the 
northern  horizon,  at  midnight ;  it  is  nearly  east,  at  6  o'clock 
in  the  morning ;  it  is  south,  at  noon,  and  about  46°  in  alti- 
tude ;  and  is  nearly  west  at  6  in  the  afternoon. 

(11.)  In  the  southern  hemisphere,  there  is  no  prominent 
dtar  near  the  south  pole  ;  that  is,  no  southern  polar  star  ;  but, 
of  course,  there  are  circumpolar  stars,  and  more  and  more  as 
one  goes  south ;  and  if  it  were  possible  to  go  to  the  south 
pole,  the  whole  southern  hemisphere  would  consist  of  circum- 
polar stars,  and  the  pole,  or  fixed  point  of  the  heavens,  would 
tc  directly  overhead ;  and  the  sun  himself,  when  south  of  the 
equator,  would  be  a  circumpolar  body,  going  round  and  round 
every  24  hours,  nearly  parallel  with  the  horizon. 

(12  )  In  all  latitudes,  and  from  all  places,  the  sun  is 


20  ASTRONOMY. 

CHAP.  i.    observed  to  circulate  round  the  nearest  pole,  as  a  center ;  and 

The  near-  when  the  sun  is  on  the  same  side  of  the  equator  as  the  ob- 

est  pole   is  server  n  ore  than  half  of  the  sun's  diurnal  circle  is  above  the 

the  center  of  ., 

the  sun's  di-  horizon,  and  the  observer  will  have  more  than  12  hours  sun- 

uraal       mo-  lig}lt. 

When  the  sun  is  on  the  equator,  the  horizon,  of  every  lati- 
tude, cuts  the  sun's  diurnal  circle  into  two  equal  parts,  and 
gives  12  hours  day,  arid  12  hours  night,  the  world  over. 
When  the  sun  is  on  the  opposite  side  of  the  equator  from  the 
observer,  the  smaller  segment  of  the  sun's  diurnal  circle  is 
above  the  horizon,  and,  of  course,  gives  shorter  days  than 
nights. 

We  have,  thus  far,  made  but  rude  arid  very  imperfect  ob- 
servations on  the  apparent  motion  of  the  heavenly  bodies,  and 
have  satisfied  ourselves  only  of  two  facts : 

FMU  »et-  1.  That  all  the  stars,  sun,  moon,  and  planets  included, 
apparently  circulate  round  the  pole,  and  round  the  earth,  in 
a  day,  or  in  about  24  hours. 

2  That  the  sun  comes  to  the  meridian,  at  different  alti- 
tudes above  the  horizon,  at  different  seasons  of  the  year, 
giving  long  days  in  June,  and  short  days  in  December. 

( 13.)  Let  us  now  pay  attention  to  some  other  particulars, 
Let  us  look  at  the  different  groups  of  stars,  and  individual 
stars,  so  that  we  can  recognize  them  night  after  night. 
Necessity      We  should  now  have  some  means  of  measuring  time ;  but, 
Vlngo*  in  early  days,  when  astronomy  was  no  further  advanced  than 
it  is  supposed  to  be  in  this  work,  a  clock  could  hardly  have 
had  existence ;  and  the  advancement  of  timepieces  has  been 
nearly  as  gradual  as  the  advancement  of  astronomy  itself. 

But  we  will  not  dwell  on  the  history,  and  difficulties,  of 
clockmaking:  whatever  these  difficulties  may  have  been,  or 
whatever  niceties  modern  science  and  art  may  have  attained, 
there  never  was  a  period  when  people  had  not  a  good  general 
idea  of  time,  and  some  means  to  measure  it.  For  /nstance, 
sunrise  and  sunset  could  be  always  noted  as  distinct  points 
of  time ;  and  the  interval  of  a  day  and  a  night,  or  an  astro- 
nomical day,  which  we  now  call  24  hours,  was  soou  observed 
to  be  a  constant  quantity. 


PRELIMINARY    OBSERVATIONS  21 

At  first,  only  rude  timepieces  could  be  made,  designed  to  CHAP.  i. 
mark  off  equal  intervals  of  time;  but  we  will  suppose,  at 
once,  that  the  reader  of  this  work,  or  our  imaginary  observer, 
can  have  the  use  of  a  common  clock,  which  measures  mean 
solar  time  of  24  hours  in  a  natural  day,  which  is  marked  by 
the  sun. 

( 14.)  Now,  having  power  to  recognize  certain  stars,  or    The  parti 
groups  of  stars,  such  as  the  Seven  Stars,  the  Belt  of  Orion,  cular  , posi' 

'  tion  of  stan 

Aldebaran,  Sirius,  and  the  like,  and  having  likewise  the  use  in  relation  to 
of  a  clock,  he  can  observe  when  any  particular  star  comes  to  time< 
any  definite  position. 

Let  a  person  place  himself  at  any  particular  point,  to  the 
north  of  any  perpendicular  line,  as  the  edge  of  a  wall  or 
building,  and  let  him  observe  the  stars  as  they  pass  behind 
the  building,  in  their  diurnal  motions  from  the  east  to  tKe 
west.  For  example,  let  us  suppose  that  the  observer  is 
watching  the  star  Aldebaran,  and  that,  when  the  eye  is  placed 
in  a  particular  definite  position,  the  star  passes  behind  the 
building  at  exactly  8  o'clock. 

The  next  evening,  the  same  star  will  come  to  the  same 
point  about  4  minutes  before  8  o'clock ;  and  it  will  not  come 
to  the  same  point  again,  at  8  o'clock  in  the  evening,  until 
after  the  expiration  of  one  year. 

(15.)  But  in  any  year,  on  the  same  day  of  the  month,  and 
at  the  same  hour  of  the  day,  the  same  star  will  be  at,  or  very 
near,  the  same  position,  as  seen  from  the  same  point. 

For  instance,  if  certain  stars  come  on  the  meridian  at  a     on    «u» 
particular  time  in  the  evening,  on  the  first  day  of  December,  oomin§r     *» 
the  same  stars  will  not  come  on  the  meridian  again,  at  the  dian. 
same  time  of  the  night,  until  the  first  day  of  the  next  December. 

On  the  first  of  January,  certain  stars  come  to  the  meridian    index    to 
at  midnight;  and  (  speaking  loosely)  every  first  of  January  thelengthof 
the  same  stars  come  to  the  meridian  at  the  same  time ;  and 
there  will  be  no  other  day  during  the  whole  year,  when  the 
same  stars  will  come  to  the  meridian  at  midnight. 

Thus,  the  same  day  of  every  year  is  observed  to  have  the 
same  position  of  the  stars  at  the  same  hour  of  the  night ;  and 
this  is  the  most  definite  index  for  the  expiration  of  a  year. 


22  ASTRONOMY. 

CHAP,  i.         (  16.)  The  year  is  also  indicated  by  the  change  of  the  sun's 

Another  declination,  which  the  most  careless  observer  cannot  fail  to 

index  of  the  notice>     Qn  the  o]  st  of  June,  the  sun  declines  about  23  1  de- 

length  of  the 

year.  grees  from  the  equator  toward  the  north  ;  and,  of  course,  to 

us  in  the  northern  hemisphere,  its  meridian  altitude  is  so 
much  greater,  and  the  horizontal  shadows  it  casts  from  the 
same  fixed  objects  will  be  shorter;  and  the  same  meridian 
altitude  and  short  shadow  will  not  occur  again  until  the  fol- 
lowing June,  or  after  the  expiration  of  one  year. 

Thus,  we  see,  that  the  time  of  the  stars  coming  on  to  the 
meridian,  and  the  declination  of  the  sun,  have  a  close  corre- 
spondence, in  relation  to  time. 

Fixed      In  all  our  observations  on  the  stars,  we  notice  that  their 
Is™  tennis  aPParen*  relative  situations  are  not  changed  by  their  diurnal 

applied.  motions.  In  whatever  parts  of  their  circles  they  are  observed, 
or  at  whatever  hour  of  the  night  they  are  seen,  the  same  con- 
figuration is  recognized,  although  the  same  group,  in  the 
different  parts  of  its  course,  will  stand  differently,  in  respect 
to  the  horizon.  For  instance,  a  configuration  of  stars  resem- 
bling the  letter  A,  when  east  of  the  meridian,  will  resemble 
the  letter  V,  when  west  of  the  meridian. 

Wander-  As  the  stars,  in  general,  do  not  change  their  positions  in 
respect  to  each  other,  they  are  called  fated  stars;  but  there 
are  a  few  important  stars  that  do  change,  in  respect  to  other 
stars;  and  for  that  reason  they  become  especial  objects  of 
attention,  and  form  the  most  interesting  portion  of  astro- 
nomy. 

jn  t^  eariiest  ages,  those  stars  that  changed  their  places, 
were  called  wandering  stars;  and  the.?  were  subsequently 
found  to  be  the  planetary  bodies  of  the  lolar  system,  like  the 
earth  on  which  we  live. 


stari* 


APPEARANCES    IN    THE    HEAVENS. 


CHAPTER  II. 

APPEARANCES  IN  THE  HEAVENS. 

IN  the  preceding  chapter  we  have  only  called  to  mind  the    CHAP.  u. 
most  obvious  and  preliminary  observations,  which  force  them- 
selves  on   every  one  who  pays  the  least  attention  to  the 
subject. 

We  shall  now  consider  the  observer  at  one  place,  making 
more  minute  and  scientific  observations. 

(17.)  We  have  already  remarked,  that  if  the  observer     HOW    to 
were  on  the  equator,  the  poles,  to  him,  would  be  in  his  horizon.  tu"de  ^ef  *h'e 
If  he  were  at  one  of  the  poles,  for  instance,  the  north  pole,  the  place  of  oi> 
equator  would  then  bound  the  horizon.     If  he  were  half  Way  serv&tion- 
between  the  equator  and  one  of  the  poles,  that  pole  would 
appear  half  way  between  the  horizon  and  the  zenith. 

Therefore,  by  observing  the  altitude  of  the  pole  above  the  hori- 
zon, we  determine  the  number  of  degrees  we  are  from  the 
equator,  which  is  called  the  latitude  of  the  place. 

(18.)  To  carry  the  mind  of  the  reader  progressively  along, 
in  astronomy,  we  must  now  suppose  that  he  not  only  has  the 
use  of  a  good  clock,  but  has  also  some  instrument  to  measure 
angles. 

Clocks  and  astronomical  instruments  progressed  toward 
perfection  in  about  the  same  ratio  as  astronomy  itself;  but, 
as  we  are  investigating  or  leading  the  young  mind  to  the  in- 
vestigation of  astronomy,  and  not  making  clocks  or  mathe- 
matical instruments,  we  therefore  suppose  that  the  observer 
has  all  the  necessary  instruments  at  his  command,  and  we 
may  now  require  him  to  make  a  correct  map  of  the  visible 
heavens ;  but  to  accomplish  it,  we  must  allow  him  at  least 
one  year's  time,  and  even  then  he  cannot  arrive  at  anything 
like  accuracy,  as  several  incidental  difficulties,  instrumental 
errors,  and  practical  inaccuracies,  must  be  met  and  overcome. 

(19.)  There  are  three  principal  sources  of  error,  which    Source.  «f 
must  be  taken  into  consideration,  in  making  astronomical 
observations.     1.  Uncertainty  as  to  the  exact  time.     2.  Inex-  tion. 


24  ASTRONOMY. 

CM^P.  fl.  pertness  and  want  of  tact  in  the  observer  ;  and  3.  Imperfec- 
tion in  the  instruments.  Everything  done  by  man  is  neces- 
sarily imperfect. 

Practical      "  It  may  be  thought  an  easy  thing,"  says  Sir  John  Her- 
scne^    "  by  one  unacquainted  with  the  niceties  required,  to 


of  error.  turn  a  circle  in  metal,  to  divide  its  circumference  into  360 
equal  parts,  and  these  again  into  smaller  subdivisions,  —  to 
place  it  accurately  on  its  center,  and  to  adjust  it  in  a  given 
position  ;  but  practically  it  is  found  to  be  one  of  the  most 
difficult.  Nor  will  this  appear  extraordinary,  when  it  is  con- 
sidered that,  owing  to  the  application  of  telescopes  to  the 
purposes  of  angular  measurement,  every  imperfection  of  struc- 
ture or  division  becomes  magnified  by  the  whole  optical  power 
of  that  instrument;  and  that  thus,  not  only  direct  errors  of 
workmanship,  arising  from  unsteadiness  of  hand  or  imperfec- 
tion of  tools,  but  those  inaccuracies  which  originate  in  far 
more  uncontrollable  causes,  such  as  the  unequal  expansion 
and  contraction  of  metallic  masses,  by  a  change  of  tempera- 
ture, and  their  unavoidable  flexure  or  bending  by  their  own 
weight,  become  perceptible  and  measurable." 

N«cessary      (  20.)  The  most  important  instruments,  in  an  observatory, 
nt8'  aside  from  the  clock,  are  a  circle,  or  sector,  for  altitudes  ;  and 
a  transit  instrument. 

The  former  consists  of  a  circle,  or  a  portion  of  a  circle,  of 
firm  and  durable  material,  divided  into  degrees,  at  the  rate 
of  360  to  the  whole  circle.  Each  degree  is  divided  into  equal 
parts;  and,  by  a  very  ingenious  mechanical  adjustment  of  an 
index,  called  a  Vernier  scale,  the  division  of  the  degree  is 
practically  (though  not  really)  subdivided  into  seconds,  or 
3600  equal  parts. 

The  whole  instrument  must  now  be  firmly  placed  and  ad- 
justed to  the  true  horizontal  (  which  is  exactly  at  right  angles 
to  a  plumb  line  ),  and  so  made  as  to  turn  in  any  direction. 
With  this  instrument  we  can  measure  angles  of  altitude. 
The  tran.  (  21.)  The  transit  instrument  is  but  a  telescope,  firmly  fas- 
tened on  a  horizontal  axis,  east  and  west,  so  that  the  telescope 
itself  moves  up  and  down  in  the  plane  of  the  meridian,  but  «an 
never  be  turned  aside  from  the  meridian  to  the  east  or 


APPEARANCES    IN    THE   HEAVENS. 


25 


Transit  Instrument. 


Meridian  Wires. 


To  place  the  instrument  in  this  posi- 
tion, is  a  very  difficult  matter ;  but  it  is 
a  difficulty  which,  at  present,  should  not 
come  under  consideration:  we  simply 
conceive  it  so  placed,  ready  for  observa- 
tions. 

"  In  the  focus  of  the  eyepiece,  and  at 
right  angles  to  the  length  of  the  tele- 
scope, is  placed  a  system  of  one  horizontal  and  five  equidis- 
tant vertical  threads  or  wires,  as  represented  in  the  annexed 
figure,  which  always  appear  in  the  field  of  view  when  properly 
illuminated,  by  day  by  the  light  of  the 
sky,  by  night  by  that  of  a  lamp,  intro- 
duced by  a  contrivance  not  necessary  here 
to  explain.  The  place  of  this  system  of 
wires  may  be  altered  by  adjusting  screws, 
giving  it  a  lateral  (horizontal)  motion; 
and  it  is  by  this  means  brought  to  such  a 
position,  that  the  middle  one  of  the  vertical  wires  shall  inter- 
sect the  line  of  collimation  of  the  telescope,  where  it  is  arrested 
and  permanently  fastened.  In  this  situation  it  is  evident 
that  the  middle  thread  will  be  a  visible  representation  of  that 
portion  of  the  celestial  meridian  to  which  the  telescope  is 
pointed ;  and  when  a  star  is  seen  to  cross  this  wire  in  the 
telescope,  it  is  in  the  act  of  culminating,  or  passing  the  celes- 
tial meridian.  The  instant  of  this  event  is  noted  by  the 
clock  or  chronometer,  which  forms  an  indispensable  accom- 
paniment of  the  transit  instrument.  For  greater  precision, 
the  moment  of  its  crossing  each  of  the  vertical  threads  is 
noted,  and  a  mean  taken,  which  (  since  the  threads  are  equi- 
distant )  would  give  exactly  the  same  result,  were  all  the 
observations  perfect,  and  will,  of  course,  tend  to  subdivide  and 
destroy  their  errors  in  an  average  of  the  whole." 

(  22.  )  Thus,  all  prepared  with  a  transit  instrument  and  a 
clock,  we  fix  on  some  bright  star,  and  mark  when  it  comes  to 
the  meridian,  or  appears  to  pass  behind  the  central  wire  of  the 
instrument.  By  noting  the  same  event  the  next  evening,  the 
next,  and  the  nexc,  we  find  the  interval  to  be  very  sensi- 


OHAP. 


A  line  in 
the  transi'. 
instrument  • 
visible  neri- 
dian. 


Practical 
artifices,  to 
attain  accu- 
racy. 


Intervals 
between  the 
fixed  stars 
passing  the 
meridian  al 
ways  con 
slant. 


26 


ASTRONOMY. 


CHAP.  II. 


bly  less  than  24  hours ;  but  the  intervals  are  equal  to  each 
other ;  and  all  the  fixed  stars  are  unanimous  in  giving  equal 
intervals  of  time  between,  two  successive  transits  of  the  same  star, 
if  measured  by  the  same  clock. 

The  following  observations  were  actually  taken  by  M. 
Arago  and  Laeroix,  in  the  small  island  of  Formentera,  in  the 
Mediterranean,  in  December,  1807. 


Date  of  Observations. 

Time  of  transit  of  the 
Star  *  Arietis. 

Intervals  between 
successive  Transits. 

1807,  Dec.  24, 

"    25, 
"    26, 

"    27, 
"    28, 

h.    m.        s. 
9    42    32.36 
9    41     29.70 
9    40    26.72 
9     39     23.90 
9     38     21.38 

h.      m.       s. 

23     58     57.34 
23     58     57.02 

23    58     57.18 
23     58     57.48 

•f    measure 
for  time. 


These  intervals  between  the  transits  agree  so  nearly,  that 
it  is  very  natural  to  suppose  them  exactly  equal,  and  the 
small  difference  of  the  fraction  of  a  second  to  arise  from  some 
slight  irregularities  of  the  clock,  or  imperfection  in  making 
the  observations. 

The  equality  of  these  intervals  is  not  only  the  same  for  all 
the  fixed  stars,  in  passing  the  meridian,  but  they  are  the 
same  in  passing  all  other  planes. 

standard  Now  as  this  has  been  the  universal  experience  of  astrono- 
mers in  all  ages,  it  completely  establishes  the  fact,  that  all 
the  fixed  stars  come  to  the  meridian  in  exactly  equal  inter- 
vals of  time ;  and  this  gives  us  a  standard  measure  for  time, 
and  the  only  standard  measure,  for  all  other  motions  are 
variable  and  unequal. 

Time  of     Again,   this  interval  must  be   the  time   that   the   earth 
the    earth's  empiovg  jn  turning  on  its  axis;  for  if  the  star  is  fixed,  it  is  a 

revolution  on  J  &  .... 

ts  axis.  mark  for  the  time  that  the  meridian  is  in  exactly  the  same 
position  in  relation  to  absolute  space. 

M.  Arago't  ^  23.)  That  the  reader  may  not  imbibe  erroneous  impres- 
sions, we  remark,  that  the  clock  used  for  the  preceding  ob- 
servations, made  by  M.  Arago  and  Lacroix,  ran  too  fast,  if  it 
was  a  common  clock,  and  too  slow,  if  it  was  an  astronomical 


APPEARANCES    IN   THE   HEAVENS.  $7 

aock.     It  was  not  mentioned  which  clock  was  used,  nor  was  CHAP.  11. 
it  material  simply  to  establish  the  fact  of  equal  intervals  ;  nor 
was  it  essential  that  the  clock  should  run  24  hours,  in  a  mean 
solar  day  :  it  was  only  essential  that  it  ran  uniformly,  and 
marked  off  equal  hours  in  equal  times. 

If  it  had  been  a  common  clock,  and  ran  at  a  perfect  rate, 
the  interval  would  have  been  23  h.  56  m.  4.09  s. 

(  24.)  In  the  preceding  section  we  have  spoken  of  an  An  astro. 
astronomical  clock.  Soon  after  the  fact  was  established  that  "J^.cm 
the  fixed  stars  came  to  the  meridian  in  equal  times,  and  that 
interval  less  than  24  hours,  astronomers  conceived  the  idea 
of  graduating  a  clock  to  that  interval,  and  dividing  it  into  24 
hours.  Thus  graduating  a  clock  to  the  stars,  and  not  to  the 
sun,  is  called  a  sidereal,  and  not  a  solar,  or  common  clock  ; 
and  as  it  was  suggested  by  astronomers,  and  used  only  for 
the  purposes  of  astronomy,  it  is  also  very  appropriately  called 
mi  astronomical  clock;  but  save  its  graduation,  and  the 
nicety  of  its  construction,  it  does  not  differ  from  a  common 
clock. 

With  a  perfect  astronomical  clock,  the  same  star  will  ^>ass  the    To   deter. 


meridian  at  exactly  the  same  time.  from  one  year's  end  to  an- 

of  an  astro- 


clock 


minetherat« 
other*     If  the  time  is  not  the  same,  the  clock  does  not  run 

_  __ 

•  Sidereal  time-has  been  slightly  modified  since  the  discovery  of  the 
precession  of  the  equinoxes,  though  such  modification  has  never  been 
distinctly  noticed  in  any  astronomical  work. 

At  first,  it  was  designed  to  graduate  the  interval  between  iwo  suc- 
cessive transits  of  the  same  star  over  the  meridian,  to  24  hours,  and  to 
call  this  a  sidereal  day  ;  which,  in  fact,  it  is. 

But  it  was  necessary,  in  some  way,  to  connect  sidereal  with  solar 
time  ;  and,  to  secure  this  end,  it  was  determined  to  commence  the  side- 
real day  (not.  from  the  passage  of  any  particular  star  across  the  meri- 
dian, but  from  the  passage  of  the  imaginary  point  in  the  heavens,  wbero 
the  sun's  path  crosses  the  vernal  equinox,  called  the  first  point  of 
Aries),  thus  making  the  sidereal  day  and  the  equinoctial  year  commence 
at  the  same  moment  of  absolute  time. 

For  some  time,  it  was  supposed  that  the  interval  between  two  suc- 
cessive transits  of  the  first  point  of  Aries,  over  the  meridian,  was  the 
same  as  two  successive  transits  of  a  star  ;  but  the  two  intervals  are  not 
identical;  the  first  point  of  Aries  has  a  very  slow  motion  westward 
among  the  stars,  which  is  called  the  precession  of  the  equinox,  and 


28  ASTRONOMY. 


ii.  to  sidereal  time;  and  the  variation  of  time,  or  the  difference 
between  the  time  when  the  star  passes  the  meridian,  and  the 
time  which  ought  to  be  shown  by  the  clock,  will  determine 
the  rate  of  the  clock.  And  with  the  rate  of  the  clock,  and  its 
error,  we  can  readily  deduce  the  true  time  from  the  time 
shown  by  the  face  of  the  clock. 

Solar  days  (  25.  )  When  we  examine  the  sun's  passage  across  the 
meridian,  and  compare  the  elapsed  intervals  with  the  sidereal 
clock,  we  find  regular  and  progressive  variations,  above  and 
below  a  mean  period,  that  cannot  be  accounted  for  by  errors 
of  observation. 

The  mean  interval,  from  one  transit  of  the  sun  to  another, 
or  from  noon  to  noon,  when  we  take  the  average  of  the  whole 
year,  is  24  hours  of  solar  time,  or  24  h.  3m.  56.5554s.  of 
sidereal  time  ;  but,  as  we  have  just  observed,  these  intervals 
are  not  uniform;  for  instance,  about  the  20th  of  December, 
they  are  about  half  a  minute  longer,  and  about  the  20th  of 
September,  they  are  as  much  shorter,  than  the  mean  period. 

The     «un      From  this  fact,  we  are  compelled  to  regard  the  sun,  not  as 
must     have  a  fjxe(j  point  ;  it  must  have  motions,  real  or  apparent,  inde- 

real  or  appa-  r  .  J  x 

rent  motion,  pendent  of  the  rotation  of  the  earth  on  its  axis. 

(  26.  )  When  we  compare  the  times  of  the  moon  passing 
the  meridian,  with  the  astronomical  clock,  we  are  very  forcibly 
struck  with  the  irregularity  of  the  interval. 

General      The  least  interval  between  two  successive  transits  of  the 
Motion     of  moon  ^  which  may  be  called  a  lunar  day  ),  is  observed  to  be 
about  24  h.  42  m.  ;  the  greatest,  25  h.  2m.  ;  and  the  mean,  or 
average,  24  h.  54m.,  of  mean  solar  time. 

These  facts  show,  conclusively,  that  the  moon  is  not  a 

which  makes  its  transits  across  the  meridian  a  fraction  of  a  second 
shorter  than  the  transits  of  a  star. 

The  time  required  for  366  transits  of  a  star  across  the  meridian,  in 
(  3".34),  three  seconds  and  thirty-four  hundredths  of  a  second  of  sidereal 
time,  greater  than  for  366  transits  of  the  equinox. 

This  difference  would  make  a  day  in  about  25000  years.  The  time 
elapsed  between  two  successive  transits  of  the  equinox  being  now 
called  a  sidereal  day  of  .....  24h.  Om.  Os.,  the 
time  between  the  transits  of  the  same  star,  is  -  24  h.  0  m.  0.00916  r 

Every  astronomer  understands  Art.  (24)  with  this  modification. 


APPEARANCES  IN  THE  HEAVENS.         29 

fixed  body,  like  a  fixed  star,  for  then  the  interval  would  be    CHAP,  it 
24  hours  of  sidereal  time. 

But  as  the  interval  is  always  more  than  24  hours,  it  shows 
that  the  general  motion  of  the  moon  is  eastward  among  the 
stars,  with  a  daily  motion  varying  from  IQi  to  16  degrees,* 
traveling,  or  appearing  to  travel,  through  the  whole  circle 
of  the  heavens  (  360°  )  in  a  little  more  than  27  days. 

Thus,  these  observations,  however  imperfectly  and  rudely     Chief  ob- 
taken,  at  once  disclose  the  important  fact,  that  the  sun  and  •'ect 
moon  are  in  constant  change  of  position,  in  relation  to  the 
stars,  and  to  each  other ;  and,  we  may  add,  that  the  chief 
object  and  study  of  astronomy,  is,  to  discover  the  reality,  the 
causes,  the  nature,  and  extent  of  such  motions. 

(  27. )  Besides  the  sun  and  moon,   several  other  bodies  Othel 

were  noticed  as  coming  to  the  meridian  at  very  unequal  in-  w" 
tervals  of  time  —  intervals  not  differing  so  much  from  24  bodies, 
sidereal  hours  as  the  moon,  but,  unlike  the  sun  and  moon, 
the  intervals  were  sometimes  more,  sometimes  less,  and  some- 
times equal  to  24  sidereal  hours. 

These  facts  show  that  these  bodies  have  a  real,  or  appa- 
rent motion,  among  the  stars,  which  is  sometimes  westward, 
sometimes  eastward,  and  sometimes  stationary ;  but,  on  the 
whole,  the  eastward  motion  preponderates ;  and,  like  the  sun 
and  moon,  they  finally  perform  revolutions  through  the  hea- 
vens from  west  to  east. 

Only  four  such  bodies  (  stars  )  were  known  to  the  ancients,    Wandering 
namely,  Venus,  Mars,  Jupiter,  and  Saturn.  8tars  ^°™ 

These  stars  are  a  portion  of  the  planets  belonging  to  our  dents, 
solar  system,  and,  by  subsequent  research,  it  was  found  that        Mode™ 
the  Earth  was  also  one  of  the  number.     As  we  come  down 
to  more  modern  times,  several  other  planets  have  been  disco- 
vered, namely,  Mercury,  Uranus,  Vesta,  Juno,  Ceres,  Pallas, 
and,  very  recently  (  1846),  the  planet  Neptune.^ 

*  Four  minutes  above  24  hours  corresponds  to  one  degree  of  arc. 

t  We  have  not  mentioned  the  names  of  these  planets  in  the  order  in 
which  they  stand  in  the  system,  but  rather  in  the  order  of  their  dis- 
covery. As  yet,  we  have  really  no  idea  of  a  planet,  or  a  planetary 
system. 


30  ASTRONOMY. 

CHAP.  n.  "We  shall  again  examine  the  meridian  passages  of  the  sun, 
moon,  and  planets,  and  deduce  other  important  facts  con- 
cerning them,  besides  that  of  their  apparent,  or  real  motions 
among  the  fixed  stars. 

observa.      (28.)  But  let  us  return  to  the  fixed  stars.     We  have 
determine  °    several  times  mentioned  the  fact,  that  the  same  star  returns 
the  meridian  to  the  same  meridian  again  and  again,  after  every  interval  of 
tii"  itar»"  °f  ^  sidereal  hours.     So  two  different  stars  come  to  the  meri- 
dian at  constant  and  invariable  intervals  of  time  from  each 
other ;  and  by  such  intervals  we  decide  how  far,  or  how  many 
degrees,  one  star  is  east  or  west  of  another.     For  instance, 
if  a  certain  fixed  star  was  observed  to  pass  the  meridian  when 
the  sidereal  clock  marked  8  hours,  and  another  star  was  ob- 
served to  pass  at  9,  just  one  sidereal  hour  after,  then  we 
know  that  the  latter  star  is  on  a  celestial  meridian,  just  15 
degrees  eastward  of  the  meridian  of  the  first  mentioned  star. 
Coireipon-  As  24  hours  corresponds  to  the  whole  circle,  360  degrees, 
>n  h  ***  t^iere^ore  one  hour  corresponds  to  15  degrees ;  and  4  minutes, 
•nd  degrew.  ia  time,  to  one  degree  of  arc.     Hence,  whatever  be  the  ob- 
served interval  of  time  between  the  passing  of  two  stars  over 
the  meridian,  that  interval  will  determine  the  actual  difference 
of  the  meridians  running  through  the  stars ;  and  when  we 
know  the  position  of  any  one,  in  relation  to  any  celestial  meri- 
dian, we  know  the  positions  of  all  whose  meridian  observations 
have  been  thus  compared. 

night   as-      The  position  of  a  star,  in  relation  to  a  particular  celestial 

**n"on'        meridian,  is  called  Right  Ascenvw,  and  may  be  expressed 

either  in  time  or  degrees.     Astronomers  have  chosen  that 


It  is  true,  we  might  mention  every  fact,  and  every  particular  re- 
specting each  planet ;  such  as  its  period  of  revolution,  size,  distance 
from  the  sun,  &c. ;  but  such  facts,  arbitrarily  stated,  would  not  convey 
the  science  of  astronomy  to  the  reader,  for  they  can  be  told  alike  to  the 
man  and  to  the  child  —  to  the  intellectual  and  to  the  dull — to  the  learned 
and  to  the  unlearned. 

To  constitute  true  knowledge  —  to  acquire  true  science  —  the  pupil 
must  not  only  know  the  fact,  but  how  that  fact  was  discovered,  or  de- 
duced from  other  facts.  Hence  we  shall  mainly  construct  our  theories 
from  observations,  as  we  pass  along,  and  teach  the  pupil  to  decide  the 
case  from  the  facts,  evidences,  and  circumstances  pr<  sented 


REFRACTION.  31 

meridian,  for  the  first  meridian,  which  passes  through  the  CHAP.  n. 
sun's  center  at  the  instant  the  sun  crosses  the  celestial  equa-  First  mert 
tor  in  the  spring,  on  the  20th  of  March.  dian- 

Right  ascension  is  measured  from  the  first  meridian,  east- 
ward, on  the  equator,  all  the  way  round  the  circle,  from  0  to 
300  degrees,  or  from  0  h.  to  24  h. 

The  reason  why  right  ascension  is  not  called  longitude  will 
be  explained  hereafter. 

(  29. )  If  we  observe  and  note  the  difference  of  sidereal    T*  find  the 
time  between  the  coming  of  a  star  to  the  meridian,  and  the  ^nlg  J^0^ 
coming  of  any  other  celestial  body,  as  the  sun,  moon,  planet,  sun,    moon, 
or  comet,  such  difference,  applied  to  the  right  ascension  of  the  an    p  a" 
star,  will  give  the  right  ascension  of  the  body. 

But  every  astronomer  regulates,  or  aims  to  regulate,  his 
sidereal  clock,  so  that  it  shall  show  Oh.  Om.  Os.  when  the 
equinox  is  on  the  meridian ;  and,  if  it  does  so,  and  runs  regu- 
larly, then  the  time  that  any  body  passes  the  meridian  by  the 
clock,  will  give  the  right  ascension  of  the  body  in  time,  with- 
out any  correction  or  calculation;  but,  practically,  this  is 
never  the  case :  a  clock  is  never  exact,  nor  can  it  ever  run 
exactly  to  any  given  rate  or  graduation. 

We  have  thus  shown  how  to  determine  the  right  ascensions 
of  the  heavenly  bodies.  We  shall  explain  how  to  find  their 
positions  in  declination,  in  the  next  chapter. 


CHAPTER   III. 

REFRACTION. POSITION    OP    THE    EQUINOX,    AND    OBLIQUITY    OP 

THE    ECLIPTIC HOW    FOUND    BY    OBSERVATION. 

(  30. )  To  determine  the  angular  distance  of  the  stars  from   CHAP.  lu 
the  pole,  the  observer  must  first  know  the  distance   of  his 
zenith  from  the  same  point. 

As  any  zenith  is  90  degrees  from  the  true  horizon,  if  the 
observer  can  find  the  altitude  of  the  pole  above  the  horizon 


ASTRONOMY 


CHAP.^III.  (  \vhich  is  the  latitude  of  the  place  of  observation  ),  he,  of 

course,  knows  the  distance  between  the  zenith  and  the  pole. 
Prepara.      ^g  fae  nortn  pOie  js  but  an  imaginary  point,  no  star  being 

tions  for  de-  '    * 

lexinining  there,  we  cannot  directly  observe  its  altitude.  But  there  is  a 
the  latitude  verv  bright  star  near  the  pole,  called  the  Polar  Star,  which, 
as  all  other  stars  in  the  same  region,  apparently  revolves 
round  the  pole,  and  comes  to  the  meridian  twice  in  24  sidereal 
hours;  once  above  the  pole,  and  once  below  it;  and  it  is 
evident  that  the  altitude  of  the  pole  itself  must  be  midway 
between  the  greatest  and  least  altitudes  of  the  same  star. 
provided  the  apparent  motion  of  the  star  round  the  pole  is  really 
in  a  circle ;  but  before  we  examine  this  fact,  we  will  show  how 
altitudes  can  be  taken  by  th\i  mural  circle.  • 

(31.)  The  mural,  or 
wall  circle,  is  a  large  me- 
tallic circle,  firmly  fas- 
tened to  a  wall,  so  that 
its  plane  shall  coincide 
with  the  plane  of  the  me- 
ridian. 

A  perpendicular  line 
through  the  center,  ZN, 
(Fig.  2),  represents  the 
zenith  and  nadir  points  ; 
and  at  right  angles  to 
this,  through  the  center, 
is  the  horizontal  line,  Hli. 

A  telescope,  Tt,  and  an  index  bar,  li,  at  right  angles  to 
•«rrve    men-  ^&  telescope,  are  firmly  fixed  together,  and  made  to  revolve 

dian        alti-  . 

on  the  center  of  the  mural  circle. 

The  circle  is  graduated  from  the  zenith  and  nadir  point? 
each  way,  to  the  horizon,  from  0  to  90  degrees. 

When  the  telescope  is  directed  to  the  horizon,  the  index 
points,  /  and  i,  will  be  at  Z  and  N,  and,  of  course,  show  0° 
of  altitude.  When  the  telescope  is  turned  perpendicular  to 
Z,  the  index  bar  will  be  horizontal,  and  indicate  90  degree* 
of  altitude. 

When  the  telescope  is  pointed  toward  any  star,  as  in  tho 


REFRACTION.  33 

figure,  the  index  points,  /and  i,  will  show  the  position  of  the  CH\P.  in. 
telescope,  or  its  angle  from  the  horizon,  which  is  the  altitude 
of  the  star. 

As  the  telescope,  and  index  of  this  instrument,  can  revolve     Mural  cu. 
freely  round  the  whole  circle,  we  can  measure  altitudes  with  ^nsiatls°  in* 
it  equally  well  from  the  north  or  the  south ;  but  as  it  turns  stmment. 
only  in  the  plane  of  the  meridian,  we  can  observe  only  meri- 
dian altitudes  with  it. 

This  instrument  has  been  called  a  transit  circle,  and,  says 
Sir  John  Herschel,  "  The  mural  circle  is,  in  fact,  at  the  same 
time,  a  transit  instrument ;  and,  if  furnished  with  a  proper 
system  of  vertical  wires  in  the  focus  of  its  telescope,  may  be 
used  as  such.  As  the  axis,  however,  is  only  supported  at  one 
end,  it  has  not  the  strength  and  permanence  necessary  for 
the  more  delicate  purposes  of  a  transit ;  nor  can  it  be  veri- 
fied, as  a  transit  may,  by  the  reversal  of  the  two  ends  of  its 
axis,  east  for  west.  .Nothing,  howover,  prevents  a  divided 
circle  being  permanently  fastened  on  the  axis  of  a  transit 
instrument,  near  to  one  of  its  extremities,  so  as  to  revolve 
with  it,  the  reading  off  being  performed  by  a  microscope 
fixed  on  one  of  its  piers.  Such  an  instrument  is  called  a 
transit  circle,  or  a  meridian  circle,  and  serves  for  the  simulta- 
neous determination  of  the  right  ascensions  and  polar  dis- 
tances of  objects  observed  with  it ;  the  time  of  transit  being 
noted  by  the  clock,  and  the  circle  being  read  off  by  the  late- 
ral microscope." 

(  32.)  To  measure  altitudes  in  all  directions,  we  must  have       A!tii»* 

\i         .  ,./,      ..  /.  ,7  .  and  azimwUi 

another  instrument,  or  a  modification  of  this.  instrument 

Conceive  this  instrument  to  turn  on  a  perpendicular  axis, 
parallel  to  Z  N,  in  place  of  being  fixed  against  a  wall ;  and 
conceive,  also,  that  the  perpendicular  axis  rests  on  the  center 
of  a  horizontal  circle,  and  on  that  circle  carries  a  horizontal 
index,  to  measure  azimuth  angles. 

This  instrument,  so  modified,  is  called  an  altitude  and  azi- 
muth instrument,  because  it  can  measure  altitudes  and  azi- 
muths at  the  same  time. 

(  33.)  After  astronomy  is  a  little  advanced,  and  the  angu- 
lar distance  of  each  particular  star,  sun,  moon,  and  planet, 
3 


34  ASTRONOMY. 

CHAP.  HI.   from  the  pole  is  known,  then  we  can  determine  the  latitude  by 

The    lati-  observing  the  meridian  altitude  of  any  known  celestial  body ; 

n<!eth  tak*n  but  before  their  positions  are  established  (  as  is  now  supposed 

tude  of  the  to  be  the  case  with  the  reader  of  this  work  ),  the  only  way  to 

pole-  observe  the  latitude  is  by  the  altitudes  of  some  circumpolar 

star,  as  mentioned  in  Art.  30. 

To  settle  this  very  important  element,  the  observer  turns 
the  telescope  of  his  mural  circle  to  the  pole  star,  and  ob- 
serves its  greatest  and  least  altitudes,  and  takes  the  half  sum 
for  his  latitude.  But  is  this  really  his  latitude  ?  Does  it 
require  any  correction,  and  if  so,  what,  and  for  what  reason? 
A  difficulty.  At  first,  it  was  very  natural  to  suppose  that  this  gave  the 
exact  latitude;  but  astronomers,  ever  suspicious,  chose  to 
verify  it,  by  taking  the  same  observations  on  other  circum- 
polar stars ;  and  if  the  theory  was  correct,  and  the  observa- 
tions correctly  taken,  all  circumpolar  stars  would  give  the 
same,  or  very  nearly  the  same,  result.  Such  observations 
were  made,  and  stars  at  the  same  distance  from  the  pole 
gave  the  same  latitude,  and  stars  at  different  distances  from 
the  pole  gave  different  latitudes ;  and  the  greater  the  dis- 
tance of  any  star  from  the  pole,  the  greater  the  latitude  de- 
duced from  it.  A  star  30  or  35  degrees  from  the  pole,  ob- 
served from  about  the  latitude  of  40  degrees,  will  give  the 
latitude  12  or  15  minutes  of  a  degree  greater  than  the  pole 
star. 

New  and  Astronomers  were  now  troubled  and  perplexed.  These 
great  and  manifest  discrepancies  could  not  be  accounted  for 
by  imperfection  of  instruments,  or  errors  of  observations,  and 
some  unconsidered  natural  cause  was  sought  for  as  a  solution. 
Curves  de-  TO  bring  more  evidence  to  bear  on  the  case,  astronomers 
examined  the  apparent  paths  of  the  stars  round  the  pole,  by 
means  of  the  altitude  and  azimuth  instrument,  and  they  were 
found  to  be  not  exact  circles ;  but  departed  more  and  more 
from  a  circle,  as  the  star  was  a  greater  and  greater  distance 
from  the  pole. 

These  curves  were  found  to  be  somewhat  like  ovals  —  the 
longer  diameter  passing  horizontally  through  the  pole  —  the 


ASTRONOMICAL    REFRACTION.  35 


upper  segments  very  nearly  semicircles,  and  the  lower  segments  UH*P.  in. 
flattened  on  their  under  sides. 

With  such  evidences  before  the  mind,  men  were  not  long 
in  deciding  that  these  discrepancies  were  owing  to 

• 

ASTRONOMICAL    REFRACTION. 

(  34.  )  It  is  shown,  in  every  treatise  on  natural  philosophy,  General 
that  light,  passing  obliquely  from  a  rarer  medium  into  a  °f  re" 
denser,  is  bent  toward  a  perpendicular  to  the  new  medium. 

Now,  when  rays  of  light  pass,  or  are  conceived  to  pass, 
from  any  celestial  object,  through  the  earth's  atmosphere  to 
an  observer,  the  rays  must  be  bent  downward,  unless  they  pass 
perpendicularly  through  the  atmosphere  ;  that  is,  come  from 
the  zenith. 


/ay>  Fig  3. 

EF,  £c     (Fig. 

3 ),  represent 
different  strata 
of  the  earth's  at- 
mosphere. Let 
8  be  a  star,  and 
conceive  a  line 
of  light  to  pass 
from  the  star 
through  the  va- 
rious strata  of 
air,  to  the  ob- 
server, at  0. 

When  it  meets  the  first  strata,  as  E  F,  it  is  slightly  bent 
downward;  and  as  the  air  becomes  more  and  more  dense,  its  increases  t! 

n        ,.  .  ,  titudes. 

retracting  power  becomes  greater  and  greater,  which  more 
and  more  bends  the  ray.  But  the  direction  of  the  ray,  at 
the  point  where  it  meets  the  eye  of  the  observer,  will  deter- 
mine the  position  of  the  star  as  seen  by  him.  Hence  the 
observer  at  0  will  see  the  star  at  s',  when  its  real  position  is 
at  s. 

As  a  ray  of  light,  from  any  celestial  object,  is  bent  down- 


36  ASTRCNOMY. 

CHAP.  ill.  ward,  therefore,  as  we  may  see  by  inspecting  the  figure,  the 
altitude  of  all  the  heavenly  bodies  is  increased  by  refraction. 

This  shows  that  all  the  altitudes,  taken  as  described  in 
Art.  33,  must,  be  apparent  altitudes  —  greater  than  true  alti- 
tudes —  and  the  resulting  latitudes,  deduced  from  them,  al1 
too  great. 

The  object  is  now  to  obtain  the  amount  of  the  refraction 
corresponding  to  the  different  altitudes,  in  order  to  correct  or 
allow  for  it. 

To  determine  the  amount  of  refraction,  we  must  resort 
to  observations  of  some  kind.     But  what  sort  of  observations 
will  meet  the  case  ? 
How  to      Conceive  an  observer  at  the  equator,  and  when  the  sun  or 

find    the     a-  . 

mount  of  re-  a  s*ar  passes  through,  or  very  near  his  zenith,  it  has  no  re- 
fraction  cor-  fraction.  But,  at  the  equator,  the  diurnal  circles  are  per- 
to'eve  '"de  Pendicular  *°  *ne  horizon ;  and  those  stars  which  are  very 
gree  of  niti-  near  the  equator,  really  change  their  altitudes  in  proportion  to 
tnde  the  time. 

Now  a  star  may  be  observed  to  pass  the  zenith,  at  the 
equator,  at  a  particular  moment :  four  hours  afterward  (  side- 
real time ),  the  zenith  distance  of  this  star  must  be  4  times  15, 
or  60  degrees,  and  its  altitude  just  30  degrees.  But,  by  ob- 
servation, the  altitude  will  be  found  to  be  30°  1'  38".  From 
this,  we  perceive,  that  1'  38"  is  the  amount  of  refraction 
corresponding  to  30  degrees  of  altitude. 

In  six  sidereal  hours  from  the  time  the  star  passed  the 

zenith,  the  true  position  of  the  star  would  be  in  the  horizon ; 

but,  by  observation,  the  altitude  would  be  33'  0",  or  a  little 

more  than  the  angular  diameter  of  the  sun. 

Amount      From  this,  we  perceive,  that  33'  0"  is  the  amount  of  re- 

of  horizontal  /,        ,.  ,    ,-•       -, 

redaction      fraction  at  the  horizon. 

Thus,  by  talcing  observations  at  all  intervals  of  time,  between 
the  zenith  and  the  horizon,  we  can  determine  the  refraction  corre- 
sponding to  every  degree  of  altitude. 

(  35.  )  In  the  last  article,  we  carried  the  observer  to  the 
equator,  to  make  the  case  clear ;  but  the  mathematician  need 
not  go  to  the  equator,  for  he  can  manage  the  case  wherever 


matlcian>8 


ASTRONOMICAL   REFRACTION.  37 

be  may  be  —  he  takes  into  consideration  the  curves,  as  men-   CHAP.  m. 
tioned  in  Art.  33. 

If  it  were  not  for  refraction,  the  curves  round  the  pole 
would  be  perfect  circles,  and  the  mathematician,  by  means  of 

z     +  method      of 

the  altitude  and  azimuth,  which  can  be  taken  at  any  and  finding    the 
every  point  of  a  curve,     can    determine  how  much  it  deviates  amount  of  re- 
from  a  circle,  and  from  thence  the  amount  of  refraction,  or 
nearly  the  amount  of  refraction,  at  the  several  points. 

By  using  the  refraction  thus  imperfectly  obtained,  he  can 
correct  his  altitudes,  and  obtain  his  latitude,  to  considerable 
accuracy.  Then,  by  repeating  his  observations,  he  can  fur- 
ther approximate  to  the  refraction. 

In  this  way,  by  a  multitude  of  observations  and  computa- 
ti  :>ns,  the  table  of  refraction  (  which  appears  among  the  tables 
of  every  astronomical  work  )  was  established  and  draVn  out. 

(  36.  )  The  effect  of  refraction,  as  we  have  already  seen,  is     Refraction 
to  increase  the  altitude  of  all  the  heavenly  bodies.     There-  ^^J*^ 
fore,  by  the  aid  of  refraction,  the  sun  rises  before  it  otherwise  light 
would,  and  does  not  set  as  soon  as  it  would  if  it  were  not 
for  refraction  ;  and  thus  the  apparent  length  of  every  day  is 
increased  by  refraction,  and  more  than  half  of  the  earth's  sur- 
face is  constantly  illuminated.     The  extra  illumination  is  equal 
to  a  zone,  entirely  round  the  earth,  of  about  40  miles  in 
breadth. 

As  the  refraction  in  the  "horizon  is  about  33'  of  a  degree, 
the  length  of  a  day,  at  the  equator,  is  more  than  four  minutes 
longer  than  it  otherwise  would  be,  and  the  nights  four  minutes 
less. 

At  all  other  places,  where  the  diurnal  circles  are  oblique 
to  the  horizon,  the  difference  is  still  greater,  especially  if  we 
take  the  average  of  the  whole  year. 

In  high  northern  latitudes,  the  long  days  of  summer  are     Effects  in 
very  materially  increased,  in  length,  by  the  effects  of  refrac-  b'gh      lati< 
tion  ;  and  near  the  pole,  the  sun  rises,  and  is  kept  above  the 
horizon,  even  for  days,  longer  than  it  otherwise  would  be, 
owing  to  the  same  cause. 

Refraction  varies  very  rapidly,  in  its  amount,  near  the  hori- 


38  ASTRONOMY.  f 

CHAP,  ui.  zon;  and  this  causes  a  visible  distortion  of  both  sun  and 

moon,  just  as  they  rise  or  set. 

D^°rt^      For  instance,  when  the  lower  limb  of  the  sun  is  just  in  the 
and  moon  in  horizon,  it  is  elevated,  by  refraction,  33'. 
the  horizon.       But  the  altitude  of  the  upper  limb  is  then  32',  and  the 
refraction,  at  this  altitude,  is  27'  50",  elevating  the  upper 
limb  by  this  quantity.     Hence,  we  perceive,  that  the  lower 
limb  is  elevated  more  than  the  upper ;  and  the  perpendicular 
diameter  of  the  sun  is  apparently  shortened  by  5'  10",  and 
the  sun  is  distinctly  seen  of  an  oval  form,    which  deviates 
more  from  a  circle  below  than  above. 

An  optical  The  apparently  dilated  size  of  the  sun  and  moon,  when 
near  the  horizon,  has  nothing  to  do  with  refraction :  it  is  a 
mere  illusion,  and  has  no  reality,  as  may  be  known  by  apply- 
ing the  following  means  of  measurement. 

Roll  up  a  tube  of  paper,  of  such  a  size  and  dimensions  as 
just  to  take  in  the  rising  moon,  at  one  end  of  the  tube,  when 
the  eye  is  at  the  other.  After  the  moon  rises  some  distance 
in  the  sky,  observe  again  with  this  tube,  and  it  will  be  found 
that  the  apparent  size  of  the  moon  will  even  more  than  fill  it. 

The  reason  of  this  illusion  is  well  understood  by  the  stu- 
dent of  philosophy;  but  we  are  now  too  much  engaged  with 
realities  to  be  drawn  aside  to  explain  illusions,  phantoms,  or 
any  Will-o'-the-wisp. 

When  small  stars  are  near  the  horizon,  they  become  invi- 
sible ;  either  the  refraction  enfeebles  and  dissipates  their  light, 
or  the  vapors,  which  are  always  floating  in  the  atmosphere, 
serve  as  a  cloud  to  obscure  them. 

Appiicatiom  (37.)  Having  shown  the  possibility  of  making  a  table  of 
refraction  corresponding  to  all  apparent  altitudes,  we  can  now, 
by  applying  its  effects  to  the  observed  altitudes  of  the  cir- 
cumpolar  stars,  obtain  the  true  latitude  of  the  place  of  obber- 
vation. 

Let  it  be  borne  in  mind,  that  the  latitude  of  any  plao.e  on 
the  earth,  is  the  inclination  of  its  zenith  to  the  plane  of  the 
equator ;  which  inclination  is  equal  to  the  altitude  of  the  pole 
above  the  horizon. 

We  demonstrate  this  as  follows.     Let  E  (  Fig.  4  )  rtpre- 


ASTRONOMICAL    REFRACTION.  3<J 

sent     the     earth.  Fig.  4.  CHAP  m 

Now,  as  an  ob- 
server always  con- 
ceives himself  to 
be  on  the  topmost 
part  of  the  earth, 
';he  vertical  point, 
-Z,  truly  and  natu- 
rally represents  his 

zenith.  Through  JE,  draw  HE  0,  at  right  angles  to  E Z\ 
then  HE  0  will  represent  the  horizon  ( for  the  horizon  is 
always  at  right  angles  to  the  zenith  ). 

Let  E  Q  represent  the  plane  of  the  equator,  and  at  right 
angles  to  it,  from  the  center  of  the  earth,  must  be  the  earth's 
axis  ;  therefore,  E  P,  at  right  angles  to  E  Q,  is  the  direction 
of  the  pole. 

Now  the  arcs,       -        -    ZP+P0=90°, 
Also,       - 


By  subtraction,      -  P  0—ZQ=Q ; 

Or,  by  transposition,  the  arc  PO  =  ZQ',  that  is,  the 
altitude  of  the  pole  is  equal  to  the  latitude  of  the  place ; 
which  was  to  be  demonstrated. 

In  the  same  manner,  we  may  demonstrate  that  the  arc 
H  Q  is  equal  to  the  arc  Z  P  ;  that  is,  the  polar  distance  of 
the  zenith  is  equal  to  the  meridian  altitude  of  the  celestial  equa- 
tor. Now,  we  perceive,  that  by  knowing  the  latitude,  we 
know  the  several  divisions  of  the  celestial  meridian,  from  the 
northern  to  the  southern  horizon,  namely,  OP,  P  Z,  Z  Q, 
and  #  H. 

(  38.)  We  are  now  prepared  to  observe  and  determine  the 
declinations  of  the  stars. 

The  declination  of  a  star,  or  any  celestial  object,  is  its  men- 
dian  distance  from  the  celestial  equator.  t1011  Jefined- 

This  corresponds  with  latitude  on  the  earth,  and  declination 
might  have  been  called  latitude. 

The  term  latitude,  as  applied  in  astronomy,  is  to  be  de- 
fined hereafter. 


40  ASTRONOMY. 

CHAP.  in.        To  determine  the  declination  of  a  star,  we  must  observe 

HOW   to  its  meridian  altitude  (  by  some  instrument,  say  the  mural 

find  the  de-    Me  ^'ig.  2  ),  and  correct  the  altitude  for  refraction  (  see 

clination  of  a 

star.  table  of  refraction  )  ;  the  difference  will  be  the  star's  true 

altitude. 

If  the  true  meridian  altitude  of  the  star  is  less  than  the  meri- 
dian altitude  of  the  celestial  equator,  then  the  declination  of  th& 
star  is  south.  If  the  meridian  altitude  of  the  star  is  greater 
than  the  meridian  altitude  of  the  equator,  then  the  decimation  of 
Hie  star  is  north. 

These  truths  will  be  apparent  by  merely  inspecting  Fig.  4. 

EXAMPLES. 

Examples  1.  Suppose  an  observer  in  the  latitude  of  40°  12'  18" 
thod'pnnwd*  nortn»  observes  the  meridian  altitude  of  a  star,  from  the 
to  find  any  southern  horizon,  to  be  31°  36'  37"  ;  what  is  the  declination 


.tar's     dec"'  Of  that  Star? 
Ration. 


From  -     90°       0'     00 

Take  the  latitude,  40       12      18 


Diff.  is  the  meridian  alt.  of  the  equator,  49°  47'     42" 
Alt.  of  star,     31°     36'     37" 

Refraction, 1 32 

True  altitude,     31°     35'       5'     -    -    31°  35' 5^ 

Decimation  of  the  star,  south,     -         -  18°  12'     37" 

2.  The  same  observer  finds  the  meridian  altitude  of  an- 
other star,  from  the  southern  horizon,  to  be  79°  31'  42"; 
what  is  the  declination  of  that  star  ? 

Observed  altitude,        -  -     79°     31'     42" 

Refraction,  -        H 

True  altitude,  -     79       31      31 
Altitude  of  equator,     ...         49      47      42 

Star's  declination,  north,  -  -     29°     43'     49" 

3.  The  same  observer,  and  from  the  same  place,  finds  the 
meridian  altitude  of  a  star,  from  the  northern  horizon,  to  bo 
5lc  29'  53";  what  is  the  declination  of  that  star? 


ASTRONOMICAL    REFRACTION.  41 

Obseived  altitude,  •  51°     29'     53"     CHAP,  in. 

Refraction,  -         -  .  46 

True  altitude  of  star,  -  51       29        7 

Altitude  of  pole  ( or  latitude ),  40       12      18 

Star  from  the  pole  (  or  polar  dist.  ),  11       16      49 

Polar  dist.,  from  90°,  gives  decl.,  north,  78°     43'     11" 

In  this  way  the  declination  of  every  star  in  the  visible 
heavens  can  be  determined. 

(  39.)  In  Art.  28  we  have  explained  how  to  obtain  the      Element! 
difference  of  the  right  ascensions  of  the  stars ;  and  in  the  last  of  the  star, 
article  we  have  shown  how  to  obtain  their  declinations. 

With  the  declinations  and  differences  of  right  ascensions,  we  may 
mark  down  the  positions  of  all  the  stars  on  a  globe  or  sphere  — 
the  true  representation,  of  the  appearance  of  the  heavens. 

Quite  a  region  of  stars  exists  around  the  south  pole,  which 
are  never  seen  from  these  northern  latitudes;  and  to  observe 
them,  and  define  their  positions,  Dr.  Halley,  Sir  John  Her- 
schel,  and  several  other  English  and  French  astronomers, 
Tiave,  at  different  periods,  visited  the  southern  hemisphere. 
Thus,  by  the  accumulated  labors  of  the  many  astronomers, 
we  at  length  have  correct  catalogues  of  all  the  stars  in  both 
hemispheres,  even  down  to  many  that  are  never  seen  by  the 
naked  eye. 

(40.)  In  Art.  28,  we  have  explained  how  to  find  the  dif-     The   zero 

.  .  L  meridian    of 

ferences  of  the  right  ascensions  of  the  stars ;  but  we  have  not  right 
yet  found  the  absolute  right  ascension  of  any  star,  for  the  want  sion> 
of  the  first  meridian,  or  zero  line,  from  which  to  reckon.  But 
astronomers  have  agreed  to  take  that  meridian  for  the  zero 
meridian,  which  passes  through  the  sun's  center  the  instant 
the  sun  comes  to  the  celestial  equator,  in  the  spring  (  which 
point  on  the  equator  is  called  the  equinoctial  point ) ;  but  the 
difficulty  is  tojind  exactly  where  (  near  what  stars  )  this  meridian 
line  is.  Before  we  can  define  this  line,  we  must  take  obser- 
vations on  the  sun,  and  determine  where  it  crosses  the  equa- 
tor, and  from  the  time  we  can  determine  the  place.  But  be- 
fore we  can  place  much  reliance  on  solar  observations,  we 
must  ask  ourselves  this  question.  Has  the  sun  any  parallax? 


42  ASTRONOMY. 

CHAP,  ill,  that  is,  is  the  position  of  the  sun  just  where  it  appears  to  be? 
Is  it  really  in  the  plane  of  the  equator,  when  it  appears  to  be 
there  ? 

Paralla*.  T;O  an  northern  observers,  is  not  the  sun  thrown  back  on  the 
face  of  the  sky,  to  a  more  southern  position  than  the  one  it 
really  occupies?  Undoubtedly  it  is;  and  this  change  of 
position,  caused  by  the  locality  of  the  observer,  is  called  paral- 
lax; but,  in  respect  to  the  sun,  it  is  too  small  to  be  considered 
in  these  primary  observations. 

The  early  astronomers  asked  themselves  these  questions, 
and  based  their  conclusions  on  the  following  consideration  : 
Sun's    pa-      If  the  sun  is  materially  projected  out  of  its  true  place  ;  if  it  is 
raiiax  msen-  thrown  to  the  southward,  as  seen  bv  a  northern  observer,  it 

eible,  in  com-      .  . 

monobserva-  will  cross  the  equator  in  the  spring  sooner  than  it  appears 

U<MIS'  to  cross. 

But  let  an  observer  be  in  the  southern  hemisphere,  and,  to 
nim,  the  sun  would  be  apparently  thrown  over  to  the  north, 
and  it  would  appear  to  cross  the  equator  before  it  really  did 
cross.  Hence,  if  the  sun  is  thrown  out.  of  place  by  parallax, 
an  observer  in  the  southern  hemisphere  would  decide  that  the 
sun  crossed  the  equator  quicker,  in  absolute  time,  than  that 
which  would  correspond  to  northern  observations. 
Northern  Buf;}  in  bringing  observations  to  the  test,  it  was  found  that 

observations  both  northern  and  southern  observers  fixed  on  the  same,  or 

compared,  very  nearly  the  same,  absolute  time  for  the  sun  crossing  the 
equator.  This  proves  that  the  position  of  the  sun  was  not 
sensibly  affected  by  parallax. 

We  will  now  suppose  ( for  the  sake  of  simplicity)  that  a 
sidereal  clock  has  been  so  regulated  as  to  run  to  the  rate  of 
sidereal  time ;  that  is,  measure  24  hours  between  any  two 
successive  transits  of  the  same  star,  over  the  same  meridian, 
but  the  sidereal  time  not  known. 

Also,  suppose  that,  at  the  Observatory  of  Greenwich,  in 
the  year  1846,  the  following  observations  were  made:* 

»  In  early  times,  such  observations  were  often  made.  We  took  these 
results  from  the  Nautical  Almanac,  and  called  them  observations  ;  but, 
for  the  purpose  of  showing  principles,  it  is  immaterial  whether  obser- 
rations  are  real  or  imaginary. 


EQUINOCTIAL   POINT, 


Date. 

Face  of  the  Side- 
rea*  Clock. 

Declination  by  Observa. 
(  Art.  38.  ) 

March  18, 
«      19, 
"      20, 
"      21, 

«      22, 

h.      m.        S. 
1       3     20.00 
1       6     58.62 
1     10     37.10 
1     14     15.47 
1     17     54.07 

o       '          " 
0     58     53.4  south, 
0     35     11.3       " 
0     11     29.4       « 
0     12     12.0  north, 
0     35     52.0       « 

43 

CHAP.  111. 

Observa- 
tions to  find 
the  equinox, 
and  the  side- 
real time. 


From  these  observations,  it  is  required  to  determine  the  sidereal 
time,  or  the  error  of  the  clock;  the  time  that  the  sun  crossed  the 
equator ;  the  sun's  right  ascension;  its  longitude,  and  the  olli- 
quity  of  the  ecliptic. 

It  is  understood  that  the  observations  for  declinations  must 
have  been  meridian  observations,  and,  of  course,  jnust  have 
been  made  at  the  instant  of  apparent  noon,  local  solar  time. 

By  merely  inspecting  these  observations,  it  will  be  perceived 
that  the  sun  must  have  crossed  the  equator  between  the  20th 
and  21st ;  for  at  the  apparent  noon  of  the  20th,  the  declina- 
tion Was  11'  29".4  south ;  and  on  the  21st,  at  apparent  noon, 
it  was  12'  12''  north.  Between  these  two  observations,  the 
clock  measured  out  24  h.  3  m.  38.37  s.,  of  sidereal  time. 

If  the  sun  had  not  changed  its  meridian  among  the  stars, 
the  time  would  have  been  just  24  hours.  The  excess 
( 3  m.  38.37  s.)  must  be  changed  into  arc,  at  the  rate  of  four 
minutes  to  one  degree.  Hence,  to  find  the  arc,  we  have  this 
proportion : 
As  4m  :  3m- 38.37s-  :  :  1°  :  to  the  required  result. 

The  result  is  54'  35". 4;  the  extent  of  arc  which  the  sun 
changed  right  ascension  during  the  interval  between  noon  and 
noon  of  the  20th  and  21st  of  March. 

To  examine  this  matter  understandingly,  draw  a  line  E  Q 
(  Fig.  5  ),  and  make  it  equal  to  54'  35".4. 

From  E,  draw  E  S  at  right  angles  to  E  Q,  and  make  it 
equal  to  11'  29".4.  From  Q,  draw  QN  at  right  angles  to 
EQ,  and  make  it  equal  to  12'  12".  Then  S  will  represent 
the  sun  at  apparent  noon,  March  20th,  and  N  the  position  of 
the  sun  at  apparent  noon  on  the  21st,  and  S Niz  the  line  o/ 


44  ASTRONOMY. 

F'g-5 the  sun  among 

the  stars,  and 
the  point  qp 
( called  the 
first  point  of 
Aries ),  and  it 
is  where  the 
sun  crosses 
the  equator. 

Now  w  e 
wish  to  find 
where  the  line 

E  Q  is  crossed  by  the  line   S  N\  or,  the  object  is,  to  find 
E  <v>,  expressed  in  time. 

To  facilitate  the  computation,  continue  E  S  to  P,  making 
SP=QN,  and  draw  the  dotted  line  P  Q.  Then  SP  Q  tf 
is  a  parallelogram.  EP=IV  29".4+12'  12"=23'  41".4; 
and  the  two  triangles,  P  E  Q  and  SE  T,  are  similar;  there- 
fore we  have 

PE  :  EQ  :  :  SE  :  E*r. 

To  have  the  value  of  E  T,  in  time,  E  Q  must  he  taken  in 
time ;  which  is  3  m.  38.37  s. 

Hence,  (23'41".4)  :  (3m-  38.37s-)  :  11'29".4  :  Ev. 

The  result  gives,     E<v=lm-  45.91s- 

But  the  clock  time  that  the  point  E  passed  the  meridian, 

was  -  -  Ih.  10  m.  37.10s. 

Add,  -  1       45.91 

Error    of      The  equi.  passed  merid.  (by  clock)  at  1  h.  12m.  23.01 

But,  at  the  instant  that  the  equinox  is  on  the  meridian, 
the  sidereal  clock  ought  to  show  Oh.  Om.  Os. 

The  error  of  the  clock  was,  therefore,  Ih.  12m.  23.01s. 
(  sub  tractive  ). 

sun's  right      As  the  whole  line,  E  Q  ( in  time  ),  is*    -     3m.  38.37  s. 
And  the  part  E  <Y>  is  -         -  1       45.91 

Therefore,  V  Q  is      -  -         -1m.  52.46 

But  qp  Q  is  the  right  ascension  of  the  sun  at  apparent  noon, 


EQUINOCTIAL    POINT.  45 

at  Greenwich,  on  the  21st  of  March,  1846;  a  very  important  CHAP.  in. 
elen?  int. 

The  right  ascension  of  any  heavenly  body,  whether  it  be       HOW   to 
sun.  moon,  star,  or  planet,  is  the  true   sidereal  time  that  it  fiml  tlie  ab 

..  solute     rig  lit 

passes  the  meridian ;  and  now,  as  we  have  the  error  of  the  ascension  ot 
clock,  we  can  determine  the  true  sidereal  time  that  any  star  the      star*. 
passes  the  meridian,  and,  of  course,  its  right  ascension  /  thus,      ,' 
for  example, 

If  a  star  passed  the  meridian  at      -     10  h.  15  m.  47  s. 

Error  of  the  clock  is  (subtractive)          1       12        23 

Eight  ascension  of  the  star  is  -       9  h.     3  m.  24  s. 

(  42.)  To  find  the  Greenwich  apparent  time,  when  the  sun 
crossed  the  equinox,  we  refer  to  Fig.  5 ;  and  as  the  point  E 
corresponds  to  apparent  noon  of  March  20th,  and  the  Q  to 
apparent  noon  of  March  21st,  and  supposing  the  motion  of 
the  sun  uniform  ( as  it  is  nearly  )  for  that  short  interval,  w< 
have  the  following  proportion : 

EQ  :  E«p  :  :  24h.  :  x.  » 

Giving  to  EQ  and  E<y  their  numeral  values  in  seconds  t»/ 
sidereal  time,  the  proportion  becomes : 

21S".37  :  105".91  :  :  24  h.    :   x. 

The  result  of  this  proportion  gives  11  h.  38m.  24s    for  cte     itme    ot, 
interval,  after  the  noon  of  the  20th  of  March,  wb.e/r  the  sun  lhe 
crossed  the  equator. 

This  result  is  in  apparent  time.  The  differeucc  between 
apparent  time,  and  mean  clock  time,  will  be  eypJained  here- 
after. At  this  period,  the  difference  between  6hti  sun  and  the 
common  clock  was  7  m.  36  s.,  to  be  added  to  apparent  time 

Equinox  of  1846,  March     -         -     20  d    11  h.  38m.  24s. 

Equation  of  time  (add),          -  7      36 

Equinox,  clock  time  (Greenwich),     20  d.  11  h.  46  m.  0 

(43.)  The  two  triangles,  JSSv>   and   ^QJV^  are  really 
spherical  triangles ;  but  triangles  on  a  sphere  whose  sides  are  of  the 
less  than  a  degree  may  be  regarded  as  plane  triangles,  with- 
out  any  appreciable  error.     In  the  triangle 


46  ASTRONOMY. 

CHAP,  in.  and,  if  we  regard  these  seconds  of  arc  as  mere  numerals,  and 
calculate  the  angle  E  T  S,  we  find  it  23°  27'  43" ;  which  in 
the  obliquity  of  tJie  ecliptic. 

Sun'i  ion-  By  computing  the  length  of  the  line  S  N,  we  find  it  59'  30"-, 
which  was  the  variation  in  the  sun's  longitude,  between  tlie  noon  of 
the  2Qth  and  21st. 

Both  longitude  and  right  ascension  are  reckoned  from  the 
equinoctial  point  <ip  :  longitude  along  the  line  qp  JV  (  which 
line  is  called  the  ecliptic),  and  right  ascension  along  the 
celestial  equator  qp  Q. 

Computing  the  length  of  the  line  qp  2it  we  find  it  equal  to 
30'  36". 6 ;  which  was  the  sun's  longitude  at  the  instant  of 
apparent  noon,  at  Greenwich,  March  21st,  1846. 
Latitude,      Meridians  of  right  ascension  are  at  right  angles  to  the  celestial 
in    astrono-  equat0r  (  at  right  angles  to  <¥  Q  ).     The  first  meridian  runs 
what     line  through  the  point  qp.     Meridians  of  latitude  are  at  right 
reckoned,      angles  to  the  ecliptic  (at  right  angles  to  the  line  SN).     La* 
titude,  in  astronomy,  is  reckoned  north  and  south  of  the  ecliptic. 

Thus  a  star  at  m  (Fig.  5),  T  n  would  be  its  longitude,  nm 
its  north  latitude ,  T  o  its  right  ascension ,  and  o  m  its  north 
declination. 

Path  of  the      (44.)  Thus,  it  may  "be  perceived,  that  these  observations 
lmu  are  very  fruitful  in  giving  important  results ;  but,  as  yet,  we 

have  used  only  two  of  them  —  those  made  on  the  20th  and  21st. 
By  bringing  the  other  observations  into  computation,  and 
extending  Fig.  5,  we  can  find  the  points  whore  the  sun  was 
on  the  other  days  mentioned ;  and  then,  by  taking  observa- 
tions every  day  in  the  year,  the  sun's  right  ascsnxion  and  lon- 
gitude can  be  determined  for  every  day,  and  its  exact  path- 
Length  of  way  through  the  apparent  celestial  sphere.  The  same  kind 
a  year,  how  Of  observations  taken  on  the  20th,  21st,  22d,  23d,  and  24th 
days  of  September,  will  show  when  the  sun  crosses  the  equa- 
tor from  north  to  south;  and  how  long  it  remains  north  of  the 
equator,  and  how  long  south  of  it.  In  March,  1847,  the 
same  observations  might  have  been  made,  and  the  exact 
length  of  an  equinoctial  year  determined :  and  in  this  way  that 
important  interval  has  been  decided,  even  to  seconds. 

The  true  length  of  an  equinoctial  year  was  early  a  very 


SOLAR    YEAR  4* 

interesting  problem  to  astronomers;  and,  before  they  had  CHAP.  in. 
good  clocks  and  refined  instruments,  it  was  one  of  some  diffi- 
culty to  settle.     But  the  more  the  difficulty,  the  greater  the 
zeal  and  perseverance ;  and  we  are  often  astonished  at  the 
accuracy  which  the  ancients  attained. 
The  length  of  the  equinoctial  year,  as  stated  in  the  tables  of 

Days,  hours,  inin.     sees 

Ptolome-e,  is         ....     365     5  55     12 

Tycho  Brahe,  made  it  365     5  48     45 

Kepler,  in  his  tables,     -         -         -     365     5  48     57 

M.  Cassini,  in  his  tables,  -  365     5  48     52 

M.  De  Lalande,    -  -     365     5  48     45 

Sir  John  lierschel,   -  365     5  48     49.7 

The  last  cannot  differ  from  the  truth  more  than  one  or  two    Soiai   «nd 

seconds.     Let  the  reader  notice  that  this  is  tn"e  equinoctial Sldereal 

year* 

year  —  the  one  that  must  ever  regulate  the  change  of  sea- 
sons. There  is  another  year  —  the  sidereal  year  —  which  is 
about  20  minutes  longer  than  the  equinoctial  year.  The  side- 
real year  is  the  time  elapsed  from  the  departure  of  the  sun 
from  the  meridian  of  ANY  STAR,  until  it  arrives  at  the  same 
meridian  again,  and  consists  of  365  d.  6  h.  9m.  9  s. 

As  the  stars  are  really  the  fixed  points  in  space,  this  latter    Cause  of 
period  is  the  apparent  revolution  of  the  sun ;  and  the  shorter  difference' 
period,  for  the  equinoctial  year,  is  caused  by  the  motion  of 
the  equinoctial  points  to  the  westward,  called  the  precession 
of  the  equinoxes.     Since  astronomers  first  began  to  record 
observations,  the  fixed  stars  have  increased,  in  right  ascension, 
about  2  hours  in  time,  or  30  degrees  of  arc. 

The  mean  annual  precession  of  the  equinoxes  is  50'M  of 
arc ;  which  will  make  a  revolution,  among  the  stars,  in  25868 
years.* 

*The  computation  is  thus:  As  50".l  is  to  the  number  of  seconds  in 
360  degrees  ;  so  is  one  year  to  the  number  of  years.  Which  gives 
25868  years,  nearly. 

We  say,  the  stars  increase  in  right  ascension  ;  and  this  is  true  ;  but 
the  stars  do  not  move  —  they  are  fixed  :  the  meridian  moves  from  the 
«tars. 


4*  ASTRONOMY. 


CHAPTER    IV. 

GEOGRAPHY  OF  THE  HEAVENS. 

CHAP,  iv.  (  45. )  THE  fixed  stars  are  the  only  landmarks  in  astrono- 
of  my,  in  respect  to  both  time  and  space.  They  seem  to  have 
been  thrown  about  in  irregular  and  ill-defined  groups  and 
clusters,  called  constellations.  The  individuals  of  these  groups 
and  clusters  differ  greatly  as  to  brightness,  hue,  and  color ; 
but  they  all  agree  in  one  attribute  —  a  high  degree  of  perma- 
nence, as  to  their  relative  positions  in  the  group;  and  the 
groups  are  as  permanent  in  respect  to  each  other.  This  has 
procured  them  the  title  of  fixed  stars ;  an  expression  which 
must  be  understood  in  a  comparative,  and  not  in  an  absolute, 
sense ;  for,  after  long  investigation,  it  is  ascertained  that 
some  of  them,  if  not  all,  are  in  motion :  although  too  slow  to 
be  perceptible,  except  by  very  delicate  observations,  conti- 
nued through  a  long  series  of  years. 

The  stars  are  also  divided  into  different  classes,  according 
•tar*/  °     e  *°  *neif  degree  of  brilliancy,  called  magnitudes.     There  are 
six  magnitudes,  visible  to  the  naked  eye ;  and  ten  telescopic 
magnitudes  —  in  all,  sixteen. 

The  brightest  are  said  to  be  of  the  first  magnitude ;  those 
less  bright,  of  the  second  magnitude,  etc. ;  the  sixth  magni- 
tude is  just  visible  to  the  naked  eye. 

One  star      The   stars  are  very  unequally  distributed  among   these 

of  the   first  classes;  nor  do  all  astronomers  agree  as  to  the  number  be- 

1  "'    longing  to  each ;  for  it  is  impossible  to  tell  where  one  class 

ends,  and  another  begins;  nor  is  it  important,  for  all  this  is 

but  a  matter  of  fancy,  involving  no  principle.     In  the  first 

magnitude  there  is  really  but  one  star  (  Sinus  ) ;  for  this  is 

manifestly  brighter  than  any  other;  but  most  astronomers 

put  15  or  20  into  this  class. 

The  second  magnitude  includes  from  50  to  60 ;  the  third, 
about  200,  the  numbers  increasing  very  rapidly,  as  we  descend 
in  the  scale  of  brightness. 

From   some  experiments  on  the  intensity  of  light,  it  has 


GEOGRAPHY  OF  THE  HEAVENS.        49 

been  determined,  that  if  we  put  the  light  of  a  star,  of  the  CHAP.  !T. 
average  1st  magnitude,  100,  we  shall  have  : 

1st  magnitude  =  100         4th  magnitude  =  6 
2d          "  =    25         5th          "         =2 

3d         "          =    12         6th          "         =1 
On  this  scale,  Sir  William  Herschel  placed  the  brightness  of 
Sirius  at  320. 

Ancient  astronomy  has  come  down  to  us  much  tarnished 
with  superstition,  and  heathen  mythology.  Every  constella- 
tion bears  the  name  of  some  pagan  deity,  and  is  associated 
with  some  absurd  and  ridiculous  fable;  yet,  strange  as  it  may 
appear,  these  masses  of  rubbish  and  ignorance  —  these  clouds 
and  fogs,  intercepting  the  true  light  of  knowledge,  are  still 
not  only  retained,  but  cherished,  in  many  modern  works,  and 
dignified  with  the  name  of  astronomy. 

Merely  as  names,  either  to  constellations  or  to  individual 


stars,  we  shall  make  no  objections;  and  it  would  be  useless,  "*mes 
if  we  did  ;  for  names  long  known,  will  be  retained,  however  nned. 
improper  or  objectionable  ;  hence,  when  we  speak  of  Orion, 
the  Little  Dog,  or  the  Great  Bear,  it  must  not  be  understood 
that  we  have  any  great  respect  for  mythology. 

It  is  not  our  purpose  now  to  describe  the  starry  heavens  — 
to  point  out  the  variable,  double,  and  multiple  stars  —  the 
Milky  Way  and  nebulae;  these  will  receive  special  attention 
in  some  future  chapter  :  at  present,  our  only  aim  is  to  point 
out  the  method  of  obtaining  a  knowledge  of  the  mere  ap- 
pearance of  the  sky,  to  the  common  observer,  which  may  be 
called  the  geography  of  the  heavens. 

To  give  a  person  an  idea  of  locality,  on  the  earth,  we  refer 
to  points  and  places  supposed  to  be  known.  Thus,  when  we 
say  that  a  certain  town  is  15  miles  north-west  of  Boston,  a 
ship  is  100  miles  east  of  the  Cape  of  Good  Hope,  or  a  cer- 
tain mountain  10  miles  north  of  Calcutta,  we  have  a  pretty 
definite  idea  of  the  localities  of  the  town,  the  ship,  and  the 
mountain,  on  the  face  of  the  earth,  provided  we  have  a  clear 
idea  of  the  face  of  the  earth,  and  know  the  position  of  Boston, 
the  Cape  of  Good  Hope,  and  Calcutta. 

So  it  is  with  the  geography  of  the  heavens;  the  apparent 
4 


50  ASTRONOMY 

CHAP.  iv.  surface  of  the  whole  heavens  must  be  in  the  mind,  and  then 
the  localities  of  certain  bright  stars  must  be  known,  as  land- 
marks, like  Boston,  the  Cape  of  Good  Hope,  and  Calcutta, 
stars  about  \ye  shan  now  make  some  effort  to  point  out  these  land- 
marks. The  North  Star  is  the  first,  and  most  important  to 
be  recognized ;  and  it  can  always  be  known  to  an  observer,  in 
any  northern  latitude,  from  its  stationary  appearance  and  alti- 
tude, equal  to  the  latitude  of  the  observer.  At  the  distance  of 
about  32  degrees  from  the  pole,  are  seven  bright  stars,  between 
the  1st  and  2d  magnitudes,  forming  a  figure  resembling  a 
dipper,  four  of  them  forming  the  cup,  and  three  the  handle.  The 
two  forming  the  sides  of  the  cup,  opposite  to  the  handle,  are 
always  in  a  line  with  the  North  Star ;  and  are  therefore  called 
pointers  :  they  ahwys  point  to  the  North  Star.  The  line  join- 
ing the  equinoxes,  or  the  first  meridian  of  right  ascension, 
runs  from  the  pole,  between  the  other  two  stars  forming  the 
cup.  The  first  star  in  the  handle,  nearest  the  cup,  is  called 
Alioth,  the  next  Mzar,  near  which  is  a  small  star,  of  the  4th 
magnitude;  the  last  one  is  Benetnasch.  The  stars  in  the 
handle  are  said  to  be  in  the  tail  of  the  Great  Bear. 

About  four  degrees  from  the  pole  star,  is  a  star  of  the  3d 
magnitude,  «  Ursce  Minoris.  A  line  drawn  through  the  pole 
(not  pole  star)  and  this  star, will  pass  through,  or  very  near, 
the  poles  of  the  ecliptic  and  the  tropics.  A  small  constella- 
tion, near  the  pole,  is  called  Ursa  Minor,  or  the  Little  Bear. 
An  irregular  semicircle  of  bright  stars,  between  the  dipper 
and  the  pole,  is  called  the  Serpent. 

imaginary  jf  a  ijne  be  drawn  from «  Ursce  Minoris,  through  the  pole 
luTto  »twm  star»  an<*  continued  about  45  degrees,  it  will  strike  a  very 
beautiful  star,  of  the  1st  magnitude,  called  Capella.  Within 
five  degrees  of  Capella  are  three  stars,  of  about  the  4th  mag- 
nitude, forming  a  very  exact  isosceles  triangle,  the  vertical 
angle  about  28  degrees.  A  line  drawn  from  Alioth,  through 
the  pole  star,  and  continued  about  the  same  distance  on  the 
other  side,  passes  through  a  cluster  of  stars  called  Cassiopia 
in  her  chair.  The  principal  star  in  Cassiopea,  with  the  pole 
star  and  Capella,  form  an  isosceles  triangle,  Capella  at  the 
vertex. 


GEOGRAPHY    OF    THE    HEAVENS. 
(46.)  More  attention  has  been  paid  to  the  constellations  CHAP. 


along  the  equator  and  ecliptic,  than  to  others  in  remoter        Ecliptic 
regions  of  the  heavens,  because  the  sun,  moon,  and  planets,  defined- 
traverse  through  them.     The  ecliptic  is  the  sun's  apparent 
annual  path  among  the  stars  (  so  called  because  all  eclipses, 
of  both  sun  and  moon,  can  take  place  only  when  the  moon  is 
either  in  or  near  this  line). 

Eight  degrees  on  each  side  of  the  ecliptic  is  called  the    signs    «,{ 
zodiac;  and  this  space  the   ancients  divided  into  12  equal  the  zodiao- 
parts  (  to  correspond  with  the  1'2  months  of  the  year  ),  and 
each  part    (  30°  )   is  called   a   sign  —  and    the   whole,    the 
signs  of  the  zodiac.     These  divisions  are  useless;  and,  of  late 
years,  astronomers  have  laid  them   aside;  yet  custom  and 
superstition  will  long  demand  a  place  for  them  in  the  common 
almanacs. 

The  signs  of  the  zodiac,  with  their  symbolic  characters,  are 
as  follows:  Aries  qp,  Taurus  tf  ,  Gemini  n,  Cancer  '03,  Leo  ft, 
Virgo  TTJ7,  Libra  =£=,  Scorpio  TT[,  Sagittarius  $  ,  Capricornus  V5>, 
Aquarius  CO-,  Pisces  X. 

Owing  to  the  precession  of  the  equinoxes,  these  signs  do 
not  correspond  with  the  constellations,  as  originally  placed  : 
the  variation  is  now  about  30  degrees;  the  stars  remain  in 
their  places  ;  and  the  first  meridian,  or  first  point  of  Aries, 
has  drawn  back,  which  has  given  to  the  stars  the  appearance 
of  moving  forward. 

Beginning  with  the  first  point  of  Aries  as  it  now  stands, 
no  prominent  star  is  near  it  ;  and,  going  along  the  ecliptic  to  facing 
the  eastward,  there  is  nothing  to  arrest  special  attention,  8 
until  we  come  to  the  Pleiades,  or  Seven  Stars,  though  only 
six  are  visible  to  the  naked  eye.  This  little  cluster  is  so  well 
known,  and  so  remarkable,  that  it  needs  no  description.  South- 
east of  the  Seven  Sfars,  at  the  distance  of  about  18  degrees, 
is  a  remarkable  cluster  of  stars,  said  to  be  in  the  Bull's  Head; 
the  largest  star  in  this  cluster  is  of  the  1st  magnitude,  of  a 
red  color,  called  Aldebaran.  It  is  one  of  the  eight  stars  se- 
lected as  points  from  which  to  compute  the  moon's  distance, 
for  the  assistance  of  navigators. 

This  cluster  resembles  an  A  when  east  of  the  meridian,  and 


52  ASTRONOMY. 


IV-  a  V  when  west  of  it.  The  Seven  Stars,  Aldebaran,  and  Ca- 
pella,  form  a  triangle  very  nearly  isosceles  —  Capella  at  the 
vertex.  A  line  drawn  from  the  Seven  Stars,  a  little  to  the 
west  of  Aldebaran,  will  strike  the  most  remarkable  constella- 
tion in  the  heavens,  Orion  (  it  is  out  of  the  zodiac,  however  )  ; 
some  call  it  the  Ell  and  Yard.  The  figure  is  mainly  distin- 
guished by  three  stars,  in  one  direction,  within  two  degrees 
of  each  other  ;  and  two  other  stars,  forming,  with  one  of  the 
three  first  mentioned,  another  line,  at  right  angles  with  the 
first  line. 

The  five  stars,  thus  in  lines,  are  of  the  1st  or  2d  magnitude. 
A  line  from  the  Seven-  Stars,  passing  near  Aldebaran  and 
through  Orion,  will  pass  very  near  to  Sirius,  the  most  bril- 
liant star  in  the  heavens.  The  ecliptic  passes  about  midway 
between  the  Seven  Stars  <tnd  Aldebaran,  in  nearly  an  eastern 
direction.  Nearly  due  east  from  the  northernmost  and  bright- 
est star  in  Orion,  and  at  the  distance  of  about  25  degrees,  is 
the  star  Procyon;  a  bright,  lone  star. 

The  northernmost  star  in  Orion,  with  Sirius  and  Procyon, 
form  an  equilateral  triangle. 

The  con-      Directly  north  of  Procyon,  at  the  distances  of  25  and  30 
•teiiatmns     Degrees,  are  two  bright  stars,  Castor  and  Pollux.     Castor  is 

are  above  the 

horizon,  and  the  most  northern.  Pollux  is  one  of  the  eight  lunar  stars. 
visible  every  Thus  we  might  run  over  that  portion  of  the  heavens  which  is 
ring  the  win-  ever  visible  to  us,  and  by  this  method  every  student  of  astro- 


ier season. 


nomy  can  render  himself  familiar  with  the  aspect  of  the  sky ; 
but  it  is  not  sufficiently  definite  and  scientific  to  satisfy  a  ma- 
thematical mind. 

( 47. )  The  only  scientific  method  of  defining  the  position 
of  a  place  on  the  earth,  is  to  mention  its  latitude  and  longitude; 
and  this  method  fully  defines  any  and  every  place,  however 
unimportant  and  unfrequented  it  may  be :  so  in  astronomy,  the 
only  scientific  methods  of  defining  the  position  of  a  star,  is  to 
mention  its  latitude  and  longitude,  or,  more  conveniently,  its 
General  ri9^  ascension  and  declination. 

and    indefi.      It  is  not  sufficient  to  tell  the  navigator  that  a  coast  makes 
mte  descnp.  Q£p  JQ  guc^  a  (jirection  from  a  certain  point,    and  that  it  is  so 

tions  not  sa- 

far  to  a  certain  cape ;  and,  from  one  cape  to  another,  it  is 


GEOGRAPHY  OF  THE  HEAVENS.        53 

about  40  miles  south-west  —  be  would  place  very  little  rcli-  CHAP.  iv. 
ance  on  any  such  directions.     To  secure  his  respect,  and     what  con« 
command  his  confidence,  the  latitude  and  longitude  of  every  «tta»te»  a  d«. 
point,  promontory,  river,  and  harbor,  along  the  coast,  must  be  scnription 
given;  and  then  he  can   shape  his  course  to  any  point,  or 
strike  in  upon  it  from  the  indefinite  expanse  of  a  pathless  sea. 
So  with  an  astronomer;  while  he  understands  and  appreciates 
the  rough  and  general  descriptions,  such  as  we  have  just  given, 
he  requires  the  certain  description,  comprised  in  right  ascension 
and  declination. 

Accordingly,  astronomers  have  given  the  right  ascensions 
and  declinations  of  every  visible  star  in  the  heavens  (  and  of 
very  many  that  are  invisible  ),  and  arranged  them  in  tables, 
in  the  order  of  right  ascension. 

There  are  far  too  many  stars,  for  each  to, -have  a  proper    John  Bay- 
name;  and,  for  the  sake  of  reference,  Mr.  John  Bayer,  of  er'8   method 

*  of  reference. 

Augsburg,  in  Suabia,  about  the  year  1603,  proposed  to  denote 
the  stars  by  the  letters  of  the  Greek  and  Roman  alphabets ; 
by  placing  the  first  Greek  letter  *  to  the  principal  star  in 
the  constellation,  &  to  the  second  in  magnitude,*  y  to  the 
third,  and  so  on;  and  if  the  Greek  alphabet  shall  become 
exhausted,  then  begin  with  the  Roman,  a,  b,  c,  etc. 

"  Catalogues  of  particular  stars,  in  sections  of  the  heavens,      Particuiw 
have  been  published  by  different  astronomers,  each  author  catal°sues- 
numbering  the  individual  stars  embraced  in  his  list,  according 
to  the  places  they  respectively  occupy  in  the  catalogue." 
These   references   to   particular   catalogues    are    sometimes 
marked  on  celestial  globes,  thus :  79  H,    meaning  that  the 
star  is  the  79th  in  Herschel's  catalogue;  37  M,  signifies  the 
37th  number  in  the  catalogue  of  Mayer,  etc. 

Among  our  tables  will  be  found  a  catalogue  of  a  hundred 
of  the  principal  stars,  inserted  for  the  purpose  of  teaching  a  defi- 
nite and  scientific  method  of  making  a  learner  acqvainted  with  the 
geography  of  the  heavens. 

To  have  a  clear  understanding  of  the  method  we  are  about 
to  explain,  we  again  consider  that  right  ascension  is  reckoned 
from  the  equinox,  eastward  along  the  equator,  from  Oh.  to 
24  hours.  When  the  sun  comes  to  the  equator,  in  March,  its 


54  ASTRONOMY. 

CHAP,  iv.  right  ascension  is  0  ;  and  from  that  time  its  right  ascension 
increases  about  four  minutes  in  a  day,  throughout  the  year, 
to  24  hours  ;  and  then  it  is  again  at  the  equinox,  and  the  *24 
hours  are  dropped. 

When  it  is  But  whatever  be  the  right  ascension  of  the  sun,  it  is  appa- 
apparent  rent  noon  when  it  comes  to  the  meridian  ;  and  the  more  east- 
ward a  body  is,  the  later  it  is  in  coming  to  the  meridian.  Thus, 
if  a  star  comes  to  the  meridian  at  two  o  clock  in  the  afternoon 
(  apparent  time  ),  it  is  because  its  right  ascension  is  TWO  HOURS 
GREATER  than  the  right  ascension  of  the  sun. 

Therefore,  if  from  the  right  ascension  of  a  star  we  subtract 
the  right  ascension  of  the  sun,  the  remainder  will  be  the  time 
for  that  star  to  come  to  the  meridian. 

Connection      ^  we  Pu*  (  ^  *  )  *°  represent  the  star's  right  ascension, 
between   R,  and  (  R  Q)  to  represent  that  of  the  sun,    and  T  to  represent 
pas-  *^e  aPParent  ^me  tnat  the  star  passes  the  meridian,  then  we 
shall  have  the  following  equation  : 


By  transposition  .  .  £*= 
That  is,  flie  rigid  ascension  of  a  star  (  or  any  celestial  body  ),  is 
equal  to  the  right  ascension  of  ilie  sun,  increased  by  the  time  thai 
the  star  (  or  body  )  comes  to  the  meridian. 

The  right  ascension  of  the  sun  is  given,  in  the  Nautical 
Almanac  (  and  in  many  other  almanacs  ),  for  every  day  in  the 
year,  when  the  sun  is  on  the  meridian  of  Greenwich;  but 
many  of  the  readers  of  this  work  may  not  have  such  an  alma- 
nac at  hand,  and,  for  their  benefit,  we  give  the  right  ascen- 
sion for  every  fifth  day  of  the  year  1846  (  Table  III  )  :  the 
local  time  is  the  apparent  noon  at  Greenwich. 

We  take  the  year  1846,  because  it  is  the  second  year  after 
leap  year  ;  and  the  sun's  right  ascension  for  any  day  in  that 
year,  will  not  differ  more  than  two  minutes  from  its  right 
ascension,  on  the  same  day,  of  any  other  year  ;  and  will  cor- 
respond with  the  right  ascension  of  the  same  day  in  1850,  by 
adding  7T\  seconds  ;  and  so  on  for  each  succeeding  period 
of  four  years. 

To  apply  the  preceding  equation,  the  observer  should  ad- 
just his  watoh  to  apparent  time  ;  that  is,  apply  the  equation 


GEOGRAPHY  OF  THE  HEAVENS.         55 

of  time,  and  know  the  direction  of  his  meridian,  at  least   CHAP,  nr 
approximately.     In  short,  by  the  range  of  definite  objects, 
he  must  be  able  to  decide,  within  two  or  three  minutes,  when  a 
celestial  body  is  on  his  meridian. 

Thus,  all  prepared,  we  will  give  a  few 

E  x  A  M  P  L  ES. 

1.    On  the  '20th  of  May  (  no  matter  what  year,  if  not  many      Exampiei 
years  from  1850),  in  the  latitude  of  40°  N.,  and  longitude  of  tofind  «*"• 
80°  W.,  at  9^.  24m.  in  ike  evening,  dock  time,  I  observed  a 
lone,  bright  star,  of  about  the  2d  magnitude,  on  the  meridian.     It 
had  a  bland,  white  light ;  and,  as  I  had  no  instrument  to  mea- 
sure its  altitude,  I  simply  judged  it  to  be  42°.      What  star 
was  it? 
We  decide  the  question  thus : 

Time  per  watch,       -         -  9h.  ''24m.     OOs. 

Equation  of  time  (see  Table),  add  3         46 

Apparent  time,  9        27         46 

Lon.  80°  W.,  equal,  in  time,  to  5        20         00 

Apparent  time,  at  Greenwich,  14        47         46 

The  right  ascension  of  the  sun,  on  the  20th  of  May  (  noon,     Correction 
Greenwich  time  ),  is  3  h.  47  m.  15  s.  (  see  Table  III).     The  of  the  sunV* 
increase,  estimated  at  the  rate  of  4  minutes  in  24  hours,  will 
give  1  minute  in  6  hours,  or  10  seconds  to  1  hour;  this,  for 
14  h.  47  m.,  gives  2  m.  27  s. 

Hence,  the  right  ascension  of  the  sun,  at  tl*  time  of  obser- 
vation, was          -         -         -         -     3  h.  49  m.  42  s. 

Apparent  time  of  observation,       -          9      27       46 

Right  ascension  of  the  star,     -         -    13  h.  17  m.  28  s. 

By  inspecting  the  catalogue  of  the  stars  (  Table  II ),  we 
find  the  right  ascension  of  Spica  to  be  13  h.  17  m.  08  s.,  and  its 
declination,  10°  21'  35".  « 

But,  in  the  latitude  of  40°  N.,  the  meridian  altitude  of  the 
celestial  equator  must  be  50° ;  and  any  stars  south  of  that- 
must  be  of  a  less  altitude.  Therefore,  the  meridian  altitude 
of  Spica  must  be  50°,  less  10°  21',  or  39°  39' ;  but  the  star 
[  observed,  I  simply  judged  to  have  had  an  altitude  of  42° 


66  ASTRONOMY. 

CHAP   iv.   It  is  very  possible  that  I  should  err,  in  altitude,  two  or  three 

degrees ;  *  but,  it  is  not  possible  that  the  star  I  obsewed  should 

be  any  other  star  than  Spica  ;  for  there  is  no  other  bright  star 

near  it.     This  is  one  of  the  lunar  stars. 

Personal      Being  now  certain  that  this  star  is  Spica,  I  can  observe  it 

observations  jn  reiatjon  ^0  j^s  appearance  —  the  small  stars  that  are  near 

recommend-    f  L 

ed  it,  and  the  clusters  of  stars  that  are  about  it  —  or  the  fact, 

that  no  remarkable  constellation  is  near  it.  In  short,  I  can 
so  make  its  acquaintance  as  to  know  it  ever  after;  but  I  am 
unable  to  convey  such  acquaintance  to  others,  by  language : 
true  knowledge,  in  this  particular,  demands  personal  obser- 
vation. 

tionofelTm'        2*     °H  ***  ^  ^  °f Ju^  1846'  ^9A'    84™'»  P'  M'>  mean 

pie«  to  find  tinw  per  watch,  a  star  of  the  \st  magnitude  came  to  the  meridian. 
*•»•  I  was  in  latitude  39°  N.,  and  about  75°  W.     The  star  was  of 

a  deep  red  color,  and,  as  near  as  my  judgment  could  decide,  its 
altitude  was  between  25°  and  30°.     Two  small  stars  were  near 
it,  and  a  remarkable  cluster  of  smaller  stars  were  west  and  north- 
west of  it,  at  the  distances  of  5°,  6°,  or  7°.      What  star  was  this  ? 
Time  per  watch,        -         -         -         -     9  h.  34  m.  00  s, 
Equa.  of  time  (  subtr.  from  mean  time  )  ,.3        48 

Apparent  time,  -         -         -     9       30        12 

Longitude,  75°,  equal  to         -         -          5 
Apparent  time,  at  Greenwich,     -         -  14  h.  30  m.  00  & 
By  examining  the  table  for  the  sun's  R.  A.,  I  find  that, 
On  the  1st  of  July,  it  is  6  h.  40  m.  00  s. 

On  the  5th,     -  6      56 30_ 

Variation,  for  4  days,  -  16  m.  30  s. 

At  this  rate,  the  variation  for  2  days,  14£  hours,  cannot  be 

»  Ten  or  twenty  degrees,  near  the  horizon,  is  apparently  a  much 
larger  space  than  the  same  number  of  degrees  near  the  zenith.  Two 
stars,  when  near  the  horizon,  appear  to  be  at  a  greater  distance  asunder 
than  when  their  altitudes  are  greater.  The  variation  is  a  mere  optical 
illusion;  for,  by  applying  instruments,  to  measure  the  angie  in  the 
different  situations,  we  find  it  the  same.  Unless  this  fact  is  taken  into 
consideration,  an  observer  will  always  conceive  the  altitude  of  any  ob- 
ject to  be  greater  than  it  really  is,  especially  if  the  altitude  is  less  than 
45  degrees. 


GEOGRAPHY  OF  THE  HEAVENS.        5* 

fai  from  10m.  10s.;  and  the  right  ascension  of  the  sun,  at   CHAP  rv 

the  time  of  observation,  must  have  been  An  exam 

Nearly  «h.  50m.  10s.  ^efigndin« 

To  which  add,  apparent  time,      -         -     9       30       12 
Right  ascension  of  the  star,     -         -        16  h.  20  m.  22  s. 
By  inspecting  the  catalogue  of  stars,  I  find  Antares  to  have 

a  right  ascension  of  16h.  20m.  2s.  and  a  declination  of  26°  4', 

south. 

In  the  latitude  mentioned,  the  meridian  altitude  of  the 

celestial  equator  must  be  50°     0' 

Objects  south  of  that  plane  must  be  less,  Jtence  (sub.)  26       4 

Meridian  altitude  of  Antares,  in  lat.  50°,  23°  56 

As  the  observation  corresponds  to  the  right  ascension  of  An- 
tares (  as  near  as  possible,  considering  errors,*in  observation, 
and  probably  in  the  watch),  and  as  the  altitudes  do  not 
differ  many  degrees  (  within  the  limits  of  guess  work  ),  it  is 
certain  that  the  star  observed  was  ANTARES.  By  its  peculiar 
red  color,  and  the  remarkable  clusters  of  stars  surrounding  it, 
I  shall  be  able  to  recognize  this  star  again,  without  the 
trouble  of  direct  observation. 

3.  On  the  night  of  the  2<M  of  June,  1846,  latitude  40°  N.,  and        TO  find 
longitude  75°  W.,  at  1  h.  48  w.  past  midnight,  clock  time,  lob-  Altair- 
served  a  star  of  the  1st  magnitude  nearly  on  the  meridian;  tw> 
other  stars,  of  about  the  3d  magnitude,  within  3°  of  it ;  the  three 
stars  forming  nearly  a  right  line,  north  and  south  ;  the  altitude 
of  the  principal  star  about  60°.      What  star  was  it? 

In  these  examples,  the  time  must  be  reckoned  on  from  noon 
to  noon  again ;  therefore  1  h.  48  m.  after  midnight  must  be 
written,  13  h.  48m.  OOs. 

Equation  of  time,  to  subtract,    -  1        12 

Apparent  time,    -         -         -         •         13       46       48 
Longitude,  ....      5 


Greenwich  apparent  time,  June  20,          18  h.   46m.  48s. 

Sun's  right  ascension,  at  this  time,  -       5  h.   57  m.  40  s. 
Time,                                              -          13      46       48 
Star's  right  ascension,      -         -  19  h.  44m.  28s. 


53  ASTRONOMY. 

CHAP.  IY.  By  inspecting  the  catalogue  of  stars,  we  find  the  right 
ascension  of  Altair  19  h.  43  m.  15  s..,  and  its  declination  8° 
27'  N.  In  latitude  40°  N.,  the  decimation  of  8°  27'  N.  will 
give  a  meridian  altitude  of  58°  27';  and,  in  short,  I  know 
the  star  observed  must  be  Altair,  and  the  two  other  stars, 
near  it,  I  recognize  in  the  catalogue. 

By  taking  these  observations,  any  person  may  become  ac- 
quainted with  all  the  principal  stars,  and  the  general  aspect 
of  the  heavens;  but  no  efforts,  confined  merely  to  the  study 
of  books,  will  accomplish  this  end. 

The  equation  in  Art.  47  is  not  confined  to  a  star ;  it  may 
be  any  heavenly  body,  moon,  comet,  or  planet.  The  time  of 
passing  the  meridian  is  but  another  term  for  right  ascension. 
If  observations  are  made  on  any  bright  star,  and  no  corre- 
sponding star  is  found  in  the  catalogue,  such  a  star  would 
probably  be  a  planet;  and  if  a  planet,  its  right  ascension 
will  change. 
Tk  6<mth.  (  48>  ^  Tlie  ^jg  regjon  Of  stars  ^th  Of  declination  50°, 

ud  Mage1  &  never  seen  in  latitude  40°  north,  nor  from  any  place  north 
.an  Ciouds  9f  that  parallel ;  and,  to  register  these  stars  in  a  catalogue,  it 
has  been  necessary  for  astronomers  to  visit  the  southern 
hemisphere,  as  we  have  before  mentioned ;  but  these  stars 
are  mostly  excluded  from  our  catalogues.  There  are  several 
constellations,  in  the  southern  region,  worthy  of  notice  —  the 
Southern  Cross  and  the  Magellan  Clouds.  The  Southern 
Cross  very  much  resembles  a  cross ;  so  much  so,  that  any 
person  would  give  the  constellation  that  appellation.  Its 
principal  star  is,  in  right  ascension,  12  h.  20  m.,  and  south 
declination  33°. 

The  Magellan  Clouds  were  at  first  supposed  to  be  clouds 
by  the  navigator  Magellan,  who  first  observed  them.  They 
are  four  in  number ;  two  are  white,  like  the  Milky  Way,  and 
have  just  the  appearance  of  little  white  clouds.  They  are 
nebula.  The  other  two  are  black  —  extremely  so  —  and  arc 
supposed  to  be  places  entirely  devoid  of  all  stars ;  yet  they 
are  in  a  very  bright  part  of  the  Milky  Way:  right  ascen- 
sion 10  h.  40  m.,  declination  62°  south. 


DESCRIPTIVE    ASTRONOMY. 

SECTION   II. 
DESCRIPTIVE   ASTRONOMY. 


CHAPTER   I. 

FIRST    CONSIDERATIONS  AS  TO  THE   DISTANCES  OF    THE  HEAVENLF 
BODIES. SIZE   AND    EXACT    FIGURE   OF    THE   EARTH. 

(  49.)  Hitherto  we  have  con-  FiS' 6' 

sidered  only  appearances,  and 
have  not  made  the  least  inquiry 
as  to  the  nature,  magnitude,  or 
distances  of  the  celestial  objects. 

Abstractly,  there  is  no  such 
thing  as  great  and  small,  near 
and  remote ;  relatively  speaking, 
however,  we  may  apply  the  terms 
great,  and  very  great,  as  regards 
both  magnitude  and  distance. 
Thus  an  error  of  ten  feet,  in  the 
measure  of  the  length  of  a 
building,  is  very  great  —  when 
an  error  of  ten  rods,  in  the  mea- 
sure of  one  hundred  miles,  would 
be  too  trifling  to  mention. 

Now  if  we  consider  the  dis- 
tance to  the  stars,  it  must  be 
relative  to  some  measure  taken 
as  a  standard,  or  our  inquiries 
will  not  be  definite,  or  even  in- 
telligible. We  now  make  this 
general  inquiry :  Are  the  heavenly  bodies  near  to,  or  remote  from, 
the  earth  ?  Here,  the  earth  itself  seems  to  be  the  natural 
standard  for  measure ;  and  if  any  body  were  but  two,  three, 
or  even  ten  times  the  diameter  of  the  earth,  in  distance,  we 


CHAP.  t. 


Are  the 
heavenly  bo- 
dies remote  t 


6tt  ASTRONOMY. 

CHAP.J.  should  call  it  near;  if  100,  200,  or  2000  times  the  diametei 
of  the  earth,  we  should  call  it  remote.  To  answer  the 
inquiry,  Are  the  heavenly  bodies  near  or  remote  ?  we  must  put 
them  to  all  possible  mathematical  tests ;  a  mere  opinion  is  of 
no  value,  without  the  foundation  of  some  positive  knowledge. 
Let  1,  2  (  Fig.  6  ),  represent  the  absolute  position  of  two 
stars ;  and  then,  if  A  B  C  represents  the  circumference  of  tho 
earth,  these  stars  may  be  said  to  be  near  ;  but  if  a  b  c  repre- 
sents the  circumference  of  the  earth,  the  stars  are  many  times 
the  diameter  of  the  earth,  in  distance,  and  therefore  may 
The  means  be  said  to  be  remote.  If  ABC  is  the  circumference  of 
this  question  tne  earfch,  in  relation  to  these  stars,  the  apparent  distance  of 
pointed  out.  the  two  stars  asunder,  as  seen  from  A,  is  measured  by  the 
angle  1 A  2 ;  and  their  apparent  distance  asunder,  as  seen 
from  the  point  B,  is  measured  by  the  angle  1  B  2 ;  and  when 
the  circumference  A  B  C  is  very  large,  as  represented  in  our 
figure,  the  angle  A,  between  the  two  stars,  is  manifestly 
greater  than  B.  But  if  a  b  c  is  the  circumference  of  the 
earth,  the  points  a  and  b  are  relatively  the  same  as  A  and  B. 
And,  it  is  an  ocular  demonstration  that  the  angle  under  which 
the  two  stars  would  appear  at  a  is  the  same,  or  nearly  the 
same,  as  that  under  which  they  would  appear  at  b;  or,  at 
least,  we  can  conceive  the  earth  so  small,  in  relation  to  the 
distance  to  the  stars,  that  the  angle  under  which  two  stars 
would  appear,  would  be  the  same  seen  from  any  point  on  the 
earth. 

The  con-  Conversely,  then,  if  the  angle  under  which  two  stars  appear 
is  the  same  as  seen  from  all  parts  of  the  earth's  surface,  it  is 
certain  that  the  diameter  of  the  earth  is  very  small,  compared 
with  the  distance  to  the  stars ;  or,  which  is  the  same  thing, 
the  distance  to  the  stars  is  many  times  the  diameter  of  the  earth. 
Therefore  observation  has  long  since  decided  this  important 
point.  Sir  John  Herschel  says :  "  The  nicest  measurements 
of  the  apparent  angular  distance  of  any  two  stars,  inter  se, 
taken  in  any  parts  of  their  diurnal  course  (  after  allowing  for 
the  unequal  effects  of  refraction,  or  when  taken  at  such  times 
that  this  cause  of  distortion  shall  act  equally  on  both  ),  mani- 
fest not  the  slightest  perceptible  variation.  Not  only  this,  but 


COMPARATIVE    DISTANCES.  61 

at  whatever  point  of  the  earth's  surface  the  measurement  is  CHAP.  L 
performed,  the  results  are  absolutely  identical.  No  instruments 
ever  yet  invented  by  man  are  delicate  enough  to  indicate,  by 
an  increase  or  diminution  of  the  angle  subtended,  that  one 
point  of  the  earth  is  nearer  to  or  farther  from  the  stars  than 
another." 

(  50.)  Perhaps  the  following  view  of  this  subject  will  be 
more  intelligible  to  the  general  reader 

Let  Z  Hjft  p.      7  distance 

//    represent  ** 8tari' 

the  celestial 
equator,  as 
seen  from  the 
equator  on 
the  earth;  and 
if  the  earth  be 
large,  in  rela- 
tion to  the 
distance  to 
the  stars,  the 
observer  will 
be  at  z' ;  and 
the  part  of  the 

celestial  arc  above  his  horizon  would  be  represented  by  A  Z  B, 
and  the  part  below  his  horizon  by  A  NB,  and  these  arcs  are  ob- 
viously unequal ;  and  their  relation  would  be  measured  by  the 
time  a  star  or  heavenly  body  remains  above  the  horizon,  com- 
pared with  the  time  below  it ;  but  by  observation  (  refraction 
being  allowed  for  ),  we  know  that  the  stars  are  as  long  above 
the  horizon  as  they  are  below;  which  shows  that  the  ob- 
server is  not  at  z',  but  at  z,  and  even  more  near  the  center ; 
so  that  the  arc  A  Z  B,  is  imperceptibly  unequal  to  the  arc  H 
NH\  that  is,  they  are  equal  to  each  other;  and  the  eartib 
is  comparatively  but  a  point,  in  relation  to  the  distance  to 
the  stars. 

This  fact  is  well  established,  as  applied  to  the  fixed  stars,     Th« 
sun,  and  planets ;  but  with  the  moon  it  is  different :  that  body  Son. 


ASTRONOMY. 

•    is  longer  below  the  horizon  than  above  it ;  which  shows  thai 
its  distance  from  the  earth  is  at  least  measurable. 

(  51.)  It  is  improper,  at  present,  or  rather,  it  is  too  advanced 
an  age,  to  pay  any  respect  to  the  ancient  notion,  that  the  earth 
is  an  extended  plane,  bounded  by  an  unknown  space,  inacces- 
sible to  men.  Common  intelligence  must  convince  even  the 
child,  that  the  earth  must  be  a  large  ball,  of  a  regular,  or  an 
irregular  shape;  for  every  one  knows  the  fact,  that  the  earth 
has  been  many  times  circumnavigated;  which  settles  the 
question. 

Eanh'»  In  addition  to  this,  any  observer  may  convince  himself,  that 
enrface  con-  the  surface  of  the  sea,  or  a  lake,  is  not  a  plane,  but  everywhere 
convex  ;  for,  in  coming  in  from  sea,  the  high  land,  back  in  the 
country,  is  seen  before  the  shore,  which  is  nearer  the  observer ; 
the  tops  of  trees,  and  the  tops  of  towers,  are  seen  before  their 
bases.  If  the  observer  is  on  shore,  viewing  an  approaching 
vessel,  he  sees  the  topmast  first ;  and  from  the  top,  downward, 
the  vessel  gradually  comes  in  view.  This  being  the  case  on 
every  sea,  and  on  every  portion  of  the  earth,  proves  that  the 
surface  of  the  earth  is  convex  on  every  part  —  hence  it  must 
be  a  globe,  or  nearly  a  globe.  These  facts,  last  mentioned, 
are  sufficiently  illustrated  by 

Fig.  8. 


(  52.)  On  the  supposition  that  the  earth  is  a  sphere,  there 

are  several  methods  of  measuring  it,  without  the  labor  of 

applying  the  measure  to  every  part  of  it.     The  first,  and 

most  natural  method  (which  we  have  already  mentioned),  is 

that  of  measuring  any  definite  portion  of  the  meridian,  and 

from  thence  computing  the  value  of  the  whole  circumference. 

HOW  to      Thus,  if  we  can  know  the  number  of  degrees,  and  parts  of 

end  the  cir-  a  Degree,  in  the  arc  A  B  (Fig.  9),  and  then  measure  the  dis- 

•fiii«  earth    tance  in  miles,  we  in  fact  virtually  know  the  whole  circumfe» 


DIAMETER    OF    THE    EARTH.  $ 

rence  ;  for  whatever  part  the  arc  A  B  is  of  360  degrees,  the    CHAP.  I. 
same  part,  the  number  of  miles  in  A  B,  is  of  the  miles  in  the 
whole  circumference. 

To  find  the  arc  A  B,  the  latitudes  of  the  two  points,  A  and 
B,  must  be  very  accurately  taken,  and  their  difference  will 
give  the  arc  in  degrees,  minutes,  and  seconds.  Now  A  B  must 
bo  measured  simply  in  distance,  as  miles,  yards,  or  feet;  but 
this  is  a  laborious  operation,  requiring  great  care  and  perse- 
verance. To  measure  directly  any  considerable  portion  of  a 
meridian,  is  indeed  impossible,  for  local  obstructions  would 
soon  compel  a  deviation  from  any  definite  line  ;  but  still  the 
measure  can  be  continued,  by  keeping  an  account  of  the  de- 
viations, and  reducing  the  measure  to  a  meridian  line. 

Let  m  be  the  miles  or  feet  in  A  B\  then  the  whole  circum- 

/  360  m  \      , 
terence  will  be  expressed  by  (  -  1~»/.  ' 

(  53.  )  When  we  know  the  Fig.  9. 

hight  of  a  mountain,  as  re- 
presented in  Fig.  9,  and  at 
the  same  time  know  the  dis- 
tance of  its  visibility  from 
the  surface  of  the  earth; 
that  is,  know  the  line  MA  ; 
then  we  can  compute  the 
line  M  C,  by  a  simple  theo- 
rem in  geometry;  thus, 


Now  as  thi3  right  hand 
member  of  this  equation  is  known,  CM  is  known  ;  and  as 
part  of  it  (  MB  )  is  already  known,  the  other  part,  B  C,  the 
diameter  of  the  earth,  thus  becomes  known. 

This  method  would  be  a  very  practical  one,  if  it  were  not      objection 
for  the  uncertainty  and  variable  nature  of  refraction  near  the  to  thu  «•• 
horizon  ;  and  for  this  reason,  this  method  is  never  relied  upon,  thod' 
although  it  often  well  agrees  with  other  methods.     As  an  ex- 
ample under  this  method,  we  give  the  following  : 


64  ASTRONOMY. 

A  mountain,  two  miles  in  perpendicular  hight,  was  seen 
from  sea  at  a  distance  of  126  miles.  If  these  data  are  cor- 
rect, what  then  is  the  diameter  of  the  earth  ? 


Solution:  MC=-==  63x126=7938.     £C= 

Dip  of  u*  (  54.  )  This  same  geometrical  theorem  serves  to  compute 
horizon.  the  dip  of  the  horizon.  The  true  horizon  is  a  right  angle  from 
the  zenith  ;  but  the  navigator,  in  consequence  of  the  motion 
of  his  vessel,  can  never  use  the  true  horizon  ;  he  must  use 
the  sea  offing,  making  allowance  for  its  dip.  If  the  naviga- 
tor's eye  were  on  a  level  with  the  sea,  and  the  sea  perfectly 
stable,  the  true  and  apparent  horizon  would  be  the  same. 
But  the  observer's  eye  must  always  be  above  the  sea  ;  and 
the  higher  it  is,  the  greater  the  dip  ;  and  the  amount  of  dip 
will  depend  on  the  hight  of  the  eye,  and  the  diameter  of  the 
earth.  The  difference  between  the  angle  AMC  (Fig.  9  ), 
and  a  right  angle  (  which  is  the  same  as  the  angle  A  EM), 
is  the  measure  of  the  dip  corresponding  to  the  hight  EM. 

For  the  benefit  of  navigators,  a  table  has  been  formed, 
showing  the  dip  for  all  common  elevations.* 

*  The  dip  is  computed  thus  : 

atlhe6  cent!  Put  EC  (Fig.  9)  =D,  BM=h  ; 

is    equal    to  /  7} 

u»  dip.          Then  EM=       +^)  ;  and  (  MA)2  = 


By  trigonometry,    (EA)*  :  (MA)2  :  :  R3  :  t&n.2AEM; 
That  is,  -     -     -      ~     i(D+h)h:i  R2  :  iim*AEM. 

For  very  moderate  elevations,  k  is  extremely  small,  in  rela- 
tion to  D  ;  and  the  second  term  of  the  proportion  may  be 
Dh.  (R  represents  the  radius  of  the  tables.)  Making  this 
consideration,  we  have 

^-  :  Dh  :  :  R*  :  iw*AEM; 

Or,  -     -      D    :    h    :  :  4Ra  :  tan.M^Jf; 
Or,  -     -    jDi  Jh.  :  :  2R  :  iim.AEM. 


DIP    OF    THE    HORIZON.  65 

(  55.  )  All  such  computations  are  made  on  the  supposition    CHAP.  L 
that  the  earth  is  exactly  spherical ;  and  it  is,  in  fact,  so  nearly 
spherical,  that  no  corrections  are  required  in  consequence  of 
its  deviation  from  that  figure. 

After  correct  views  began  to  be  entertained,  as  to  the  mag-     The  earth 
nitude  of  the  earth,  and  its  revolution  on  an  axis,  philosophers  not   e*actly 
concluded  that  its  equatorial  diameter  might  be  greater  than  "P 
its  polar  diameter;  and  investigations  have  been  made  to 
decide  the  fact. 

If  the  earth  were  exactly  spherical,  it  is  plain  that  the  cui- 
vature  over  its  surface  would  be  the  same  in  every  latitude ; 
but  if  not  of  that  figure,  a  degree  would  be  longer  on  one  part 
of  the  earth  than  on  another.  "  But,"  says  Herschel,  "when 
•we  come  to  compare  the  measures  of  meridional  arcs  made  in 
various  parts  of  the  globe,  the  results  obtained,  although  they 
agree  sufficiently  to  show  that  the  supposition  of  a  spherical 
figure  is  not  very  remote  from  the  truth,  yet  exhibit  discord- 
ances far  greater  than  what  we  have  shown  to  be  attributable 
to  error  of  observation;  and  which  render  it  evident  that  the 
hypothesis,  in  strictness  of  its  wording,  is  untenable.  The 
following  table  exhibits  the  lengths  of  a  degree  of  the  meri- 
dian (  astronomically  determined  as  above  described),  ex- 


By  inspecting  this  last  proportion,  it  will  be  perceived  that 
the  tangent  of  the  dip  varies  as  the  square  root  of  the  eleva- 
tion. To  apply  this  proportion,  we  adduce  the  following 
problem : 

The  diameter  of  the  earth  is  7912  miles ;  the  elevation  of 
the  eye,  above  the  surface,  is  ten  feet.  What  is  the  dip? 

2J2  .  .  log. 10.301030 

N/F.  .  log. _:500000 

Product  of  the  means  (log.),    ....    10.801030 
I)  mile*,  7912,  -     -     log.  -  3.898286 
Feet,  -     5280,  -     -     log.  -  3.722634 
2)^620920 

in  feet,   -     -     (log.)        3.810460  .  .  8_810460_ 
tan.  3'  22"     -     -     -    6.990570 


66 


ASTRONOMY. 


CHAP.  r.  pressed  in  British  standard  feet,  as  resulting  from  actual 
measurement,  made  with  all  possible  care  and  precision,  by 
commissioners  of  various  nations,  men  of  the  first  eminence, 
supplied  by  their  respective  governments  with  the  best  instru- 
ments, and  furnished  with  every  facility  which  could  tend  to 
insure  a  successful  result  of  their  important  labors. 


Country. 

Latitude 
of  Middle  of 
the  Arc. 

.              iLength  of 

*&u  |coDn:f;edeed 

Observers. 

Sweden     .... 

66  20  10 
58  17  37 
52  35  45 
4652    2 
4451    2 
4259    0 
39  12    0 
33  18  30 
16    822 
12  32  21 
1  31    0 

1°37'19" 
3  35     5 
3  57  13 
8  20     0 
12  22  13 
2    9  47 
1  28  45 
1  13  174 
15  57  40 
1  34  56 
373 

365782 
365368 
364971 
364872 
364535 
364262 
363786 
364713 
363044 
363013 
362808 

Svanberg. 
Struve. 
Roy,  Kater. 
Lacaille,  Cassini. 
Delambre.  Mechain. 
Boscovich. 
Mason,  Dixon. 
Lacaille. 
Lambton,  Everest. 
Lambton. 
Condamine,  etc. 

Knp'land  

France 

Rome  

America,  U.  S-.  . 
Cap«  of  G.  Hope 
India  

India    .        ... 

Peru  

The  earth 


<;  It  is  evident,  from  a  mere  inspection  of  the  second  and 
!  fourth  columns  of  this  table,  that  the  measured  length  of  a  de- 

A&n  at.  the  gree  increases  with  the  latitude,  being  greatest  near  the  poles, 

**nator         and  least  near  the  equator." 

"Assuming,"  continues  Herschel,  "that  the  earth  is  an 
ellipse,  the  geometrical  properties  of  that  figure  enable  us  to 
assign  the  proportion  between  the  lengths  of  its  axes  which 
shall  correspond  to  any  proposed  rate  of  variation  in  its  cur- 
vature, as  well  as  to  fix  upon  their  absolute  lengths,  corre- 
sponding to  any  assigned  length  of  the  degree  in  a  given 
latitude.  Without  troubling  the  reader  with  the  investiga- 
tion (which  may  be  found  in  any  work  on  the  conic  sections), 
it  will  be  sufficient  to  state  that  the  lengths,  which  agree  on 
the  whole  best  with  the  entire  series  of  meridional  arcs,  which 
have  been  satisfactorily  measured,  are  as  follow  :  — 

Feet.  Miles. 

Greater,  or  equatorial  diam.,  =41,847,426=7925.643 
Lesser,  or  polar  diam.,  -  -  =41,707,620=7899.170 
Difference  of  diameters,  or  _  jgg  806=  26  478 

polar  compression,  -     -     - 
The  proportion  of  the  diameters  is  very  nearly  that  of 


FORM    OF    THE    EARTH.  6-r 

298  :  299,  and  their  difference  ^g-  of  the  greater,  or  a  very    CHA»  f. 
little  greater  than  3-^-." 

(  5f).  )  The  shape  of  the  earth,  thus  ascertained  by  actual 
measurement,  is  just  what  theory  would  give  to  a  body  of 
water  equal  to  our  globe,  and  revolving  on  an  axis  in  24 
Lours;  and  this  has  caused  many  philosophers  to  suppose  that 
the  earth  was  formerly  in  a  fluid  state. 

If  the  earth  were  a  sphere,  a  plumb  line  at  any  point  on       Expiana. 
its  surface  would  tend  directly  toward  the  center  of  gravity  ^ion°-fradiul 
of  the  body ;  but  the  earth  being  an  ellipsoid,  or  an  oblate 
splieroid,  and  the  plumb  lines,  being  perpendicular  to  the  sur- 
face at  any  point,  do  not  tend  to  the  center  of  gravity  of  the 
figure,  but  to  points  as  represented  in  Fig.  10. 

The  plumb  line  at  H  tends  to 
F,  yet  the  mathematical  center, 
and  center  of  gravity  of  the 
figure,  is  at  E.  So  at  I,  the 
plumb  line  tends  to  the  point  (?; 
and  as  the  length  of  a  degree  at 
A,  is  to  the  length  of  a  degree 
at  H,  so  is  16?  to  II F.  If, 
however,  a  passage  were  made 
through  the  earth,  and  a  body  let  drop  through  it,  the  body 
would  not  pass  from  /to  G:  its  first  tendency  at  /would  be 
toward  the  point  G\  but  after  it  passed  below  the  surface  at 
I,  its  tendency  would  be  more  and  more  toward  the  point  E, 
the  center  of  gravity ;  but  it  would  not  pass  exactly  through 
that  point,  unless  dropped  from  the  point  A,  or  the  point  C. 

(  57.  )  If  the  earth  were  a  perfect  and  stationary  sphere,     Force    ot 
the  force  of  gravity,  on  its  surface,  would  be  everywhere  the  gravity  difie- 

•,,.,.,...,  ,.  -,  rent  on  difle- 

same ;  but,  it  being  neither  stationary,  nor  a  perfect  sphere,  renl  parts  of 
the  force  of  gravity,  on  the  different  parts  of  its  surface,  must  the     eartb 
be  different.     The  points  on  its  surface  nearest  its  center  of 
gravity,  must  have  more  attraction  than  other  points  more 
remote  from  the  center  of  gravity ;  and  if  those  points  which 
are  more  remote  from  the  center  of  gravity  have  also  a  rotary 
motion,  there  will  be  a  diminution  of  gravity  on  that  account. 
Let  .4  2?  (Fig.  10)  represent  the  equatorial  diameter  of 


68 


ASTRONOMY. 


CHAP-  !-  the  earth,  and  CD  the  polar  diameter;  and  it  is  obvious 
that  E  will  be  the  center  of  gravity,  of  the  whole  figure,  and 
Gravity  di-  t^at  t]ie  force  Of  gravity  at  C  and  D  will  be  greater  than  at 
rotation.  ^  anJ  ot^er  points  on  the  surface,  because  E  C,  or  ED,  are 
less  than  any  other  lines  from  the  point  E  to  the  surface. 
The  force  of  gravity  will  be  greatest  on  the  points  C  and  D, 
aiso,  because  they  are  stationary :  all  other  points  are  in  a 
circular  motion ;  and  circular  motion  has  a  tendency  to  depart 
from  the  center  of  motion,  and,  of  course,  to  diminish  gravity. 
The  diminution  of  the  earth's  gravity  by  the  rotation  on  it? 
axis,  amounts  to  its  ¥| »  part,*  at  the  equator.  By  this  frao- 


Computa. 
lion  of  the 
amount  of 
diminution. 


Fiff.  11 


*  Let  D  be  the  equatorial  diametei 
of  the  earth,  F  the  versed  sine  of  an  arc 
corresponding  to  the  motion  in  a  second 
of  time,  and  c  the  chord  or  arc  (  for  the 
chord  and  arc  of  so  small  a  portion  of  tho 
circumference  will  coincide,  practically 
speaking). 

A  portion  of  the  earth's  gravity,  equal 
to  Ft  is  destroyed  by  the  rotation  of  the  earth,  and  we  aro 
now  to  compute  its  value. 

By  proportional  triangles,  F  :  c  :  :  c  :  D; 

c3 

The  value  of  c  is  found  by  dividing  the  whole  circumference 
into  as  many  equal  parts  as  there  are  seconds  in  the  time  of 
revolution.  But  the  time  of  revolution  is  23  h.  56  m.  4  s.,  =? 
86164  seconds. 

The  whole  circumference  is  (3.1416)2>; 

(3.1416)1? 
"(86164)" 
(3.1416)21) 


Therefore, 


•     (2) 


By  this  value  of  c,  we  have      F—- 


(86164)2 

The  visible  force  of  gravity,  at  the  equator,  is  the  distance 
a  body  will  fall  the  first  second  of  time,  expressed  in  feet. 
Let  us  call  this  distance  g.  Now  the  part  of  gravity  des- 


EFFECT  OF  FORM  ON  GRAVITY.        69 

tion,  then,  is  the  weight  of  the  sea  about  the  equator  lightened, 
and  thereby  rendered  susceptible  of  being  supported  at  a 
higher  level  than  at  the  poles,  where  no  such  counteracting 
force  exists. 


fcroyed  by  rotation,  as  we  have  just  seen,  is  -=  ;  therefore  tho 


whole  force  of  gravity  is  (g-\--jr  ) 


Our  next  inquiry  is:  what  part  of  the  whole  is  the  part  de-   Ratio  of  *• 

diminutiot 


ttnyedf     Or  what  part  of  O+  is       ?  3°mputea' 

Which,  by  common  arithmetic,  is, 


(86164)* 
~  (3.1416)2/>  : 

Hence, 

ffl>=  (86164)^  (86164)^(16.07) 

c2  ="  (3.1416)2D~(3.1416)2(7925)(5280)' 
By  the  application  of  logarithms,  we  soon  find  the  value  of 

1 

this  expression  to  be  288.4.     Therefore,    gD       =-         . 

— +1     289.4 

We  may  now  inquire,  how  rapidly  the  earth  must  revolve 
on  its  axis,  so  that  the  whole  of  gravity  would  be  destroyed 
on  the  equator.  That  is,  so  that  F  shall  equal  g.  Equation 


(1)  then  becomes,  <7—-fi,  or  c= 

But  as  often  as  c  is  contained  in  the  whole  circumference, 
is  the  corresponding  number  of  seconds  in  a  revolution;  thai 
is,  the  time  in  seconds  must  correspond  to  the  expression, 


70  ASTRONOMY. 

CHAf-  *•  (  58. )  It  is  this  centrifugal  force  itself  that  changed  the 
shape  of  the  earth,  and  made  the  equatorial  diameter  greater 
than  the  polar.  Here,  then,  we  have  the  same  cause,  exer- 
cising at  once  a  direct  and  an  indirect  influence.  The  amount 
Rotation  Of  the  former  (  as  we  may  see  by  the  note  )  is  easily  calcu- 
md  indirect  ^ated ;  that  of  the  latter  is  far  more  difficult,  and  requires  a 
effect  on  gra-  knowledge  of  the  integral  calculus,  "  But  it  has  been  clearly 
treated  by  Newton,  Maclaurin,  Clairaut,  and  many  other  emi- 
nent geometers ;  and  the  result  of  their  investigations  is  to 
show,  that  owing  to  the  elliptic  form  of  the  earth  alone,  and 
independently  of  the  centrifugal  force,  its  attraction  ought  to 
increase  the  weight  of  a  body,  in  going  from  the  equator  to 
the  pole,  by  nearly  its  j^ th  part;  which,  together  with  the 
^|¥  th  part,  due  from  centrifugal  force,  make  the  whole  quan- 
tity 7ir4- th  part;  which  corresponds  with  observations  as 
deduced  from  the  vibrations  of  pendulums." —  See  Natural 
Philosophy. 

(  59. )  The  form  of  the  earth 
is  so  nearly  a  sphere,  that  it  is 
considered  such,  in  geography, 
navigation,  and  in  the  general 
problems  of  astronomy. 

The  average  length  of  a  de- 
Srcc  *s  ^'U  ^ng^sn  miles ;  and, 
as  this  number  is  fractional,  ar.d 
inconvenient,  navigators  have  ta- 
citly agreed  to  retain  the  ancient, 

rough  estimate  of  sixty  miles  to  a  degree ;  calling  the  mile  a 
geographical  mile.  Therefore,  the  geographical  mile  is  longer 
than  the  English  mile. 

D,  in  feet,  =  (7925)(5280) ;  g  =  16.076.  By  the  applica- 
tion of  logarithms,  we  find  this  expression  to  be  5069  seconds, 
or  Ih.  24m.  29  s.;  which  is  about  17  times  the  rapidity  of 
its  present  rotation. 

In  a  subsequent  portion  of  this  work,  we  shall  show  how 
to  arrive  at  this  result  by  another  principle,  and  through 
another  operation. 


CONVERGENCY    OF   MERIDIANS.  71 

As  all  meridians  come  together  at  the  pole,  it  follows  that    CHAP.  I. 
a  degree,  between  the  meridians,  will  become  less  and  less  as 
we  approach  the  pole ;  and  it  is  an  interesting  problem  to 
trace  the  law  of  decrease.* 


*  This  law  of  decrease  will  become  apparent,  by  inspecting 
Fig.  12.  Let  EQ  represent  a  degree,  on  the  equator,  and 
EQC  a  sector  on  the  plane  of  the  equator,  and  of  course  EC 
is  at  right  angles  to  the  axis  C  P.  Let  D  ^/be  any  plane 
parallel  to  EQC\  then  we  shall  have  the  following  proportion : 

EG    :     DI    :  :     EQ     :     DF. 

In  trigonometry,  E  C  is  known  as  the  radius  of  the  sphere; 
D  /as  the  cosine  of  the  latitude  of  the  point  D  (the  nume- 
rical values  of  sines  and  cosines,  of  all  arcs,  are  given  in  trigo- 
nometrical tables)  :  therefore  we  have  the , following  rule,  to 
compute  the  length  of  a  degree  between  two  meridians,  on 
any  parallel  of  latitude. 

RULE.  —  As  radius  is  to  the  cosine  of  the  latitude /  so  is  the 
length  of  a  degree  on  the  equator,  to  the  length  of  a  parallel  de- 
gree in  that  latitude. 

Calling  a  degree,  on  the  equator,  60  miles,  what  is  the 
length  of  a  degree  of  longitude,  in  latitude  42°  ? 

SOLUTION   BY    LOGARITHMS. 

As  radius  (see  tables),  10.000000 

Is  to  cosine  42°  (see  tables),  -                  -      9.871073 

So  is  60  miles  (log.),  -  1.778151 

To  44r\V_  mfleSj   .         .  .      1.649224 

Ai  the  latitude  of  60°,  the  degree  of  longitude  is  30  miles; 
the  diminution  is  very  slow  near  the  equator,  and  very  rapid 
near  the  poles. 

In  navigation,  the  DJ<"s  are  the  known  quantities  ob-     TO 
tained  by  the  estimations  from  the  log  line,  etc. ;  and  the  denarture  to 
navigator  wishes  to  convert   them  into  longitude,  or,  what 
is  the  same  thing,  he  wishes  to  find  their  values  projected  on 
the  equator,  and  he  states  the  proportion  thus : 
DI    :    EC    :  :     DF    :    EQ ; 

That  is,  as  cosine  of  latitude  is  to  radius,  so  is  departure  to 
difference  of  longitude. 


ASTRONOMY. 


CHAPTER   II. 


PARALLAX,  GENERAL  AND  HORIZONTAL.  —  RELATION  BETWEEN 
PARALLAX  AND  DISTANCE. REAL  DIAMETER  AND  MAGNI- 
TUDE OF  TILE  MOON. 


CHAP    U. 


(  60.  )  PARALLAX  is  a  subject  of  very  great  importance  in 
astronomy :  it  is  the  key  to  the  measure  of  the  planets  —  to 
their  distances  from  the  earth  —  and  to  the  magnitude  of  the 
whole  solar  system. 

Parallax  in      Parallax  is  tJie  difference  in  position,  of  any  body,  as  seen 
from  the  center  of  the  earth,  and  from  its  surface. 

When  a  body  is  in  the  zenith  of  any  observer,  to  him  it  hag 
no  parallax;  for  he  sees  it  in  the  same  place  in  the  heavens, 
as  though  he  viewed  it  from  the  center  of  the  earth.  The 
greatest  possible  parallax  that  a  body  can  have,  takes  place 
when  the  body  is  in  the  horizon  of  the  observer ;  and  this 
parallax  is  called  horizontal  parallax.  Hereafter,  when  wo 
speak  of  the  parallax  of  a  body,  horizontal  parallax  is  to  be 
understood,  unless  otherwise  expressed. 

A  clear  and  summary  illustration  of  parallax  in  general,  is 
given  by  Fig.  13. 
Horizontal  FUr.  13.  Let    (7  be 

parallax.  ^•»ra^H>«BMiKi*wwp*s^Hm^l^^^»*i<ra!KWBRB9m     - 

™  the  center  of 
the  earth,  Z 
the  observer, 
and  P,  or  p, 
the  position 
of  a  body. 
From  the 
center  of  the 
earth,  the 
body  is  seen 
in  the  direc- 
tion of  the 
line  C  P,  or  Cp\  from  the  observer  at  Z,  it  is  seen  in  the 


PARALLAX.  "3 

direction  of  Z  P,  or  Zp ;  and  the  difference  in  direction,  of    CHAP.  n. 
these  two  lines,  is  parallax.     When  P  is  in  the  zenith,  there 
is  no  parallax ;  when  P  is  in  the  horizon,  the  angle  Z  P  C  is 
then  greatest,  and  is  the  horizontal  parallax. 

We  now  perceive  that  the  horizontal  parallax  of  any  body       Relation 
is  equal  to  the  apparent  semidiamefer  of  the  earth,  as  seen  from  between  P»- 
the  body.     The  greater  the  distance  to  the  body,  the  less  the  distance, 
horizontal  parallax ;  and  when  the  distance  is  so  great  that 
the  semidiameter  of  the  earth  would  appear  only  as  a  point, 
then  the  body  has  no  parallax.     Conversely,  if  we  can  detect 
no  sensible  parallax,  we  know  that  the  body  must  be  at  a 
vast  distance  from  the  earth ,  and  the  earth  itself  appear  as 
a  point  from  such  a  body,  if,  in  fact,  it  were  even  visible. 

Trigonometry  gives  the  relation  between  the  angles  and 
sides  of  every  conceivable  triangle ;  therefore  we  know  all 
about  the  horizontal  triangle  Z  C  P,  when' we  know  CZ  and 
the  angles.  Calling  the  horizontal  parallax  of  any  bodyj^, 
and  the  radius  of  the  earth  r,  and  the  distance  of  the  body 
from  the  center  of  the  earth  x  ( the  radius  of  the  table  always 
R,  or  unity},  then,  by  trigonometry,  we  have, 

R    :     x     :  :     sin. p    :    r: 

(n 

sm.jt? 

From  this  equation  we  have  the  following  general  ruie,  to 
find  the  distance  to  any  celestial  body : 

RULE.  —  Divide  the  radius  of  the  tables  by  the  sine  of  the  RnJe  te 
horizontal  parallax.  Multiply  that  quotient  by  the  semidiamefer  find  the  *'»• 
of  the  earth,  and  the  product  will  be  the  result.  *ances  J°  *"• 

This  result  will,  of  course,  be  in  the  same  terms  of  linear  bodies 
measure  as  the  semidiameter  of  the  earth :  that  is,  if  r  is  in 
feet,  the  result  will  be  in  feet ;  if  r  is  in  miles,  the  result  will 
be  in  miles,  etc.  :  but,  for  astronomy,  our  terrestrial  measures 
are  too  diminutive,  to  be  convenient  (not  to  say  inappropri- 
ate) ;  and,  for  this  reason,  it  is  customary  to  call  the  semidia- 
meter of  the  earth  unity ,  and  then  the  distance  of  any  body 
from  the  earth  is  simply  the  quotient  arising  from  dividing 
il*e  radius  by  the  sine  of  tJie  horizontal  parallax  pertaining  to 


74  ASTRONOMY. 


ii.  the  body  ;  and  it  is  obvious,  that  the  less  the  parallax,  the 
greater  this  quotient;  that  is,  the  greater  the  distance  to  the 
body;  and  the  difficulty,  and  the  only  difficulty,  is  to  obtain 
the  horizontal  parallax. 

Horizontal      (61.)  The  horizontal  parallax  cannot  be  directly  observed, 

not*  b«  "b"  ^7  reason  of  the  great  amount  and  irregularity  of  horizontal 

««nr»d.         refraction  ;  but  if  we  can  obtain  a  parallax  at  any  considera- 

ble altitude,  we  can  compute  the  horizontal  parallax  there- 

from.* 

The  fixed  stars  have  no  sensible  horizontal  parallax,  as  we 
have  frequently  mentioned  ;  and  the  parallax  of  the  sun  is 
so  small,  that  it  cannot  be  directly  observed  (  see  40  )  ;  the 
moon  is  the  only  celestial  body  that  comes  forward  and  pre- 
sents its  parallax  ;  and  from  thence  we  know  that  the  moon 
is  the  only  body  that  is  within  a  moderate  distance  of  the 
earth. 

That  the  moon  had  a  sensible  parallax,  was  known  to  the 
earliest  observers,  even  before  mathematical  instruments  were 
at  all  refined;  but,  to  decide  upon  its  exact  amount,  and 
detect  its  variations,  required  the  combined  knowledge  and 
observations  of  modern  astronomers. 


induction  *  In  the  two  triangles  Zp  C  and  ZP  C  (Fig.  13),  call  the 
an&le  P  the  Parallax  in  altitude,  and  the  angle  ZPC=x, 
and  Cp  and  C P  each  equal  D.  Then,  by  trigonometry, 
we  have 

sin.  pZC    :     sin.j?      ::     D    :    r\ 
And    -     -         R         :    sin.  x     :  :     D    :    r. 
Therefore,  by  equality  of  ratios  (see  algebra), 
sin.  pZC    :     sin.jt)      :  :     R    :     sin.  a?. 

But  the  sine  pZC  is  the  sine  of  the  apparent  zenith  dis- 
tance. Therefore, 

R  sin.  p 

sin.  x=— :-, — =^ ; 

sin.  zenith  distance 

That  is ,  the  sine  of  the  hor''z<>nlcl  parallax  is  equal  to  the  sine 
of  the  parallax  in  altitude ',  into  the  radius,  and  divided  by  the 
tine  of  tfo  apparent  zenith  distance. 


LUNAR    PARALLAX  7 

The  lunar  parallax  was  first  recognized  in  northern  Europe  C"*P-  »- 
by  the  moon  appearing  to  describe  more  than  a  semicircle  south  By 
of  the  equator,  and  less  than  a  semicircle  north  of  that  line; 
and,  on  an  average,  it  was  observed  to  be  a  longer  time 
south,  than  north  of  the  equator  ;  but  no  such  inequality  could  fij*1  lndlc 
be  observed  from  the  region  of  the  equator, 

Observers  at  the  south  of  the  equator,  observing  the  posi- 
tion of  the  moon,  see  it  for  a  longer  time  north  of  the  equator 
than  south  of  it  ;  on-  ',  to  than,  it  appears  to  describe  more  t/ian 
a  semicircle  no  th  of  the  equator. 

Here,  then,  we  have  observation  against  observation,  unless 
we  can  reconcile  them.  But  the  only  reconciliation  that  can 
be  made,  is  to  conclude  that  the  moon  is  really  as  long  in  one 
hemisphere  as  the  other,  and  the  observed  discrepancy  must 
arise  from  the  positions  of  the  observers  ;  an,d  when  we  reflect 
that  parallax  must  always  depress  the  object  (  see  Fig.  13  ), 
and  throw  it  farther  from  the  observer,  it  is  therefore  per- 
fectly clear  that  a  northern  observer  should  see  the  moon 
farther  to  the  south  than  it  really  is  ,  and  a  southern  observer 
sec  the  same  body  farther  north  than  its  true  position. 

(  62.)  To  find  the  amount  of  the  lunar  parallax,  requires 
the  concurrence  of  two  observers.  They  should  be  near  the 
same  meridian,  and  as  far  apart,  in  respect  to  latitude,  as 
possible  ;  and  every  circumstance,  that  could  affect  the  result, 
must  be  known. 

The  two  most  favorable  stations  are  Greenwich  (England) 
and  the  Cape  of  Good  Hope.     They  would  be  more  favorable 
if  they  were  on  the  same  meridian  ;  but  the  small  change  in  mount  ot  •> 
declination,  while  the  moon  is  passing  from  one  meridian  to  '* 
the  other,  can  be  allowed  for  ;  and  thus  the  two  observations 
are  reduced  to  the  same  meridian,  and  equivalent  to  being 
made  at  the  same  time. 

The  most  favorable  times  for  such  observations,  are  when 
the  moon  is  near  her  greatest  declinations,  for  then  the  change 
of  declination  is  extremely  slow. 

Let  A  (  Fig.  14  )  represent  the  place  of  the  Greenwich  ob- 
servatory, and  B  the  station  at  the  Cape  of  Good  Hope. 
C  is  the  center  of  the  earth,  and  Z  and  Z'  are  the  zenith 


jjjj" 


ASTRONOMY. 


OBAP.JI.  Fig.  14.  points  of  the  observers.  Let  M 

be  the  position  of  the  moon,  and 
the  observer  at  A  will  see  it  pro- 
jected on  the  sky  at  m',  and  tho 
observer  at  B  will  see  it  pro- 
jected on  the  sky  at  m. 

Now  the  figure  A  CBM  is  a 
quadrilateral;  the  angle  A  C  B 
is  known  by  the  latitudes  of  the 
two  observers;  the  angles  MA 
C  and  MB  C  are  the  respective 
zenith  distances,  taken  from  180°. 

But  the  sum  of  all  the  angles 
of  any  quadrilateral  is  equal  to 
four  right  angles  ;  and  hence  the 
angles  at  A,  C,  and  £,  being 
known,  the  parallactic  angle  at 
M  is  known. 

In  this  quadrilateral,  then,  we 
have  two  sides,  A  C  and  C  B% 
land  all  the  angles;  and  this  is 
sufficient  for  the  most  ordinary 
mathematician  to  decide  every 
particular  in  connection  with  it  ; 
that  is,  we  can  find  AM,  M  j?, 
and  finally  MC.*  Now  MO  being  known,  the  horizontal 

A  mam..      *  The  direct  and  analytical  method  of  obtaining  M  C,  will  be 
1   de"  very  acceptable  to  the  young  mathematician  ;  and,  for  that 
reason,  we  give  it. 

Put  AC=C£=r,  CM=x,  and  the  two  parts  of  the  ob- 
served parallactic  angle,  M,  represented  by  P  and  Q,  as  in 
the  figure.     Also,  let  a  represent  the  natural  sine  of  the  angle 
MAC,  and  b  the  natural  sine  of  the  angle  MB  C: 
Then,  by  trigonometry,  -     x  :  a  :  :  r  :  sin.  Q  ; 
Also,       -     -     -     -     -     -  x  :  b  :  : 


doc  t  ion. 


sn. 


Hence, 


sin.  P+sin.  Q= 


.  (I) 


J.UNAR    PARALLAX.  77 

parallax  can  be  computed,  for  it  is  but  a.  function  of  the  dls-    CHA».  n 
tance  (see  60). 

By  the  equation  (Art.  GO),  #=(  -. jr 


(72    \ 
—  Jr;  and  whenar, 


the 


distance,  is  known,  sin.  p,  or  sine  of  the  horizontal  parallax, 
is  known. 

(  63.  )  The  result  of  such  observations,  taken  at  different  Variable 
times,  show  all  values  to  MC,  between  55^,  and  63TY0- ;  n™  to 
taking  the  value  of  r  as  unity. 

These  variations  are  regular  and  systematic,  both  as  to 
time  and  place,  in  the  heavens ;  and  they  show,  without  fur- 
ther investigation,  that  the  moon  does  not  go  round  the  earth 
in  a  circle,  or,  if  it  does,  the  earth  is  not  in  the  center  of  that 
circle. 

The  parallaxes  corresponding  to  these  extreme  distances, 
are  61'  29"  and  53'  50". 

When  the  moon  moves  round  to  that  part  of  her  orbit 
which  is  most  remote  from  the  earth,  it  is  said  to  be  in  apogee;  and 
and,  when  nearest  to  the  earth,  it  is  said  to  be  in  perigee. 
The  points  apogee  and  perigee,  mainly  opposite  to  each  other, 
do  not  keep  the  same  places  in  the  heavens,  but  gradually 
move  forward  in  the  same  direction  as  the  motion  of  the  moon, 
and  perform  a  revolution  in  a  little  less  than  nine  years. 


But,  by  a  general  theorem  in  trigonometry, 

P.L.Q  P Q 

sin.  P+  sin.  Q=^2  sin.— ^  cos.— -A     .      (2) 

Now  by  equating  (1)  and  (2),  and  observing  that  P-^-Q=* 

(p Q\ 
cos. — ^ — )  must  be  extremely  near  unity; 

and,  therefore,  as  a  factor,  may  disappear ;  we  then  have, 

0    .    M      (a+ft> 
2  sm.-5-  =  v    '    '  ,     or     x=- 
2  a?  2  sin. 

A  more  ancient  method  is  to  compute  the  value  of  the  little 
eriangle  B  C  G,  and  then  of  the  whole  triangle  A  MO,  and 
then  of  a  part  A  MC  or  M  Q  C. 


78  ASTRONOMY. 

OHAP.  n.  (64.)  Many  times,  when  the  moon  comes  round  to  its  peri- 
gee, we  find  its  parallax  less  than  61'  29",  and,  at  the  oppo- 
site apogee,  more  than  53'  50".  It  is  only  when  the  sun  is 
in,  or  near  a  line  with  the  lunar  perigee  and  apogee,  that 
these  greatest  extremes  are  observed  to  happen  ;  and  when 
the  sun  is  near  a  right  angle  to  the  perigee  and  apogee,  then 
the  moon  moves  round  the  earth  in  an  orbit  nearer  a  circle  ; 
and  thus,  by  observing  with  care  the  variation  of  the  moon's 
parallax,  we  find  that  its  orbit  is  a  revolving  ellipse,  of  variable 
eccentricity. 

(65.)  Because  the  moon's  distance  from  the  earth  is  va- 
riable, therefore  there  must  be  a  mean  distance:  we  shall 
show,  hereafter,  that  her  motion  is  variable  ;  therefore  there 
is  a  mean  motion  ;  and,  as  the  eccentricity  is  variable,  there 
is  a  mean  eccentricity. 

MBA*  Pa.  The  extreme  parallaxes,  at  mean  eccentricity,  are  60'  20" 
parallax  at  an<^  ^'  ^"  >  an(^  *ne  corresponding  distances  from  the  earth 
MEAN  dis-  are  56.93  and  63.  04,  the  radius  of  the  earth  being  unity. 


mean  paral]aXj  or  mean  between  60'  20"  and  54'  05",  is 
57'  12".5;  but  the  parallax,  at  mean  distance,  is  57'  03"*. 

*  It  may  seem  paradoxical  that  the  mean  parallax,  and  the 
parallax  at  mean  distance  arc  different  quantities  ;  but  the 
following  investigation  will  set  the  matter  at  rest.  Let  d  and 
D  be  extreme  distances,  and  M  the  mean  distance. 

Then,    -     ---     rf+D=2Jf.     .         .         .         (1) 

Also,  let  p  and  P  be  the  parallaxes  corresponding  to  the  dis- 
tances d  and  D  ;  and  put  x  to  represent  the  parallax  at  mean 
distance.  Then,  by  Art.  60  (  if  we  call  the  radius  of  the 
tables  unity),  we  have 

i  i 

and  M=- 


sm.  x 
Substituting  these  values  of  d,  D,  and  M,  in  equation  (1)  w« 

1  1  2 

have,    -     -      -. \-  -. -  =  —. —  ; 

sm.jp    '    sin.  P        sin.* 

_                                                       2  sin.  p  sin.  P    X0x 
Or,  ...     sin.  P  +  sin.  p   = (2) 


VARIATION    OF    PARALLAX.  79 

.    55.92+63-84                  CH*P.  n. 
The  mean  between  extreme  distances  is ^ or  o9.oo;     

but  the  true  mean  distance  is  60.26,  corresponding  to  the     Mean  du- 
parallax  57'  3".     The  mean,  between  extremes,  is  a  variable  m>on 
quantity;  but  the   true  mean  distance  is  ever  the  same,  a 
little  more  than  60i  times  the  semidiameter  of  the  earth. 

(  66.)  The  variations  in  the  moon's  real  distance  must  cor- 
respond to  apparent  variations  in  the  moon's  diameter ;  and  if 
the  moon,  or  any  other  body,  should  have  no  variation  in 
apparent  diameter,  we  should  then  conclude  that  the  body 
was  always  at  the  same  distance  from  us. 

The  change,  in  apparent  diameter,  of  any  heavenly  body,  is 
numerically  proportioned  to  its  real  change  in  distance;  as 
appears  from  the  demonstration  in  the  note  below.* 

But  by  a  well  known,  and  general  theorem  in  trigonometry,     Mean  p* 

rallax. 


(P-L-p\  /P n\ 

— 3r4-;  J  cos.  (  •          )    (3) 
A      /  \     'A      ' 

By  equating  (3)  and  (2),  and  observing  that  the  cosines 
of  very  small  arcs  may  be  practically  taken  as  unity,  or  ra- 
dius ;  therefore, 

'P-\-p\     _  sin.  P  sin.  p 


sin. 

sm.  x 


_  .  sin.  P  sin.  p 

Or,  .....    sm.  x  = 


On  applying  this  equation,  we  find  #=57'  3  . 

*  Let  A  be  the  _  Fig-  15- 
point  of  vision,  and 
d  the  diameter  of  | 
any  body  at  diffe- 


40. 

Now,  by  trigonometry,  we  nave  the  following  proportions  : 
AC  :    d    ::    H,    :    tan.  CAD 
AB  :    d    :  :    R    :     tan.  BAE. 


80  A  3  T  R  0  N  O  M  Y. 

CHAP  ii.  Now  if  the  moon  has  a  real  change  in  distance,  as  observa- 
tions show,  such  change  must  be  accompanied  with  apparent 
changes  in  the  moon's  diameter;  and,  by  directing  observa- 
tions to  this  particular,  we  find  a  perfect  correspondence; 
showing  the  harmony  of  truth,  and  the  beauties  of  real 
science. 

Connec-      We  have  several  times  mentioned  that  the  moon's  horizon- 
tion  between  ^j  para}]ax  [s  ^Q  gemidiameter  of  the  earth,  as  seen  from  the 

ieruiliame- 

ter  and  hon-  moon ;  and  now  we  further  say,  that  what  we  call  the  moon's 
rental  parai-  gemidiameter,  an  observer  at  the  moon  would  call  the  earth's 
horizontal  parallax ;  and  the  variation  of  these  two  angles  de- 
pends on  the  same  circumstance  —  the  variation  of  the  distance 
between  the  earth  and  moon;  and,  depending  on  one  and  the 
same  cause,  they  must  vary  in  just  the  same  proportion. 

When  the  moon's  horizontal  parallax  is  greatest,  the  moon's 
semidiameter  is  greatest;  and,  when  least,  the  semidiameter 
is  the  least ;  and  if  we  divide  the  tangent  of  the  semidiameter 
by  the  tangent  of  its  horizontal  parallax,  we  shall  always  find 
the  same  quotient  (the  decimal  0.27293) ;  and  that  quotient 
is  the  ratio  between  the  real  diameter  of  the  earth  and  the 
diameter  of  the  moon.*  Having  this  ratio,  and  the  diameter 
of  the  earth,  7912  miles,  we  can  compute  the  diameter  of  the 

moon  thus : 

7912x0.27293=2169.4  miles. 

From  the  first  proportion,  -     -     -   A  C  tan.  CAD=dR ; 

From  the  second, AB  tan.  BAE=dR : 

By  equality,  -     -     -     -  A  Ctan.  CAD=AB  tan.  BAE. 

This  last  equation,  put  into  an  equivalent  proportion,  gives : 
AC    :    AB    :    tan.  BAE    ::     tan.  CAD. 

But  tangents  of  very  small  arcs  (  such  as  those  under  which 

the  heavenly  bodies  appear)  are  to  each  other  as  the  arcs 

themselves.     Therefore, 

AC    :    AB    :  :     angle  BAE    :     angle  CAD; 

That  is ;  the  angular  measures  of  the  same  body  are  inversely 

proportional  to  the  corresponding  distances. 

*  This  requires  demonstration.     Let  E  be  the  real  semi- 


APPEARANCE    FROM    THE    MOON  8l 

As  spheres  are  to  each  other  in  proportion  to  the  cults  of    CHAP.  IT, 
their  diameters,  therefore  the  bulk  (not  mass)  of  the  earth. 
is  to  that  of  the  moon,  as  1  to  •£•$,  nearly. 

A.c<  the  moon's  distance  is  60  j-  times  the  radius  of  the  earth,       Angmen- 
it  follows  that  it  is  about  rVtn  nearer  to  us,  when  at  the  talion,  of  the 

°  u  ^  moon's  semi. 

zenith,  than  when  in  the  horizon.     Making  allowance  for  this  diameter  :  iu 
(  in  proportion  to  the    sine    of  the  altitude  ),  is  called  the  cause> 
augmentation  of  the  semidiameter. 

(  68.  )  It  may  be  remarked,  by  every  one,  that  we  always     The  earth 

see  the  same  face  of  the  moon  ;  which  shows  that  she  must  a  moon   to 

•    •  •  -i  *^e  m»°* 

roil  on  an  axis  in  the  same  time  as  her  mean  revolution  about 

the  earth  ;  for,  if  she  kept  her  surface  toward  the  same  part 
of  the  heavens,  it  could  not  be  constantly  presented  to  the 
earth,  because,  to  her  view,  the  earth  revolves  round  the 
moon,  the  same  as  to  us  the  moon  revolv9S  round  the  earth  ; 
and  the  earth  presents  phases  to  the  moon,  as  the  moon  does 
to  us.  except  opposite  in  time,  because  the  two  bodies  are 
opposite  in  position.  When  we  have  new  moon,  the  lunarians 
have  full  earth  ;  and  when  -we  have  first  quarter,  they  have 
last  quarter,  etc.  The  moon  appears,  to  .us,  about  half  a 
degree  in  diameter  ;  the  earth  appears,  to  them,  a  moon,  about 

diameter  of 

the    earth 

(Fig.l6),iw 

that  of  the 

moon,  D  the 

distance  be- 

tween  the 

two  bodies  ;  and  let  the  radius  of  the  tables  be  unity.     Put 

P  to  represent  the  moon's  horizontal  parallax,  and  *  its  appa- 

rent semidiameter.     Then,  by  trigonometry, 

D  :  E  :  :  1  :  tan.  P;    and   D  :  m  :  :  1  :  tan.  * 


From  the  first,  2)=—— ^;  from  the  2d,  J9= 


vn 


tan.  s 


nn      f  Em  tan.*       m 

Therefore,- -  =  - ,    or b= --, .  Q.  E<  D 

tan.  P      tan.  *  tan.  P      E 


82  ASTRONOMY 

CHAP.  ii.    two  degrees  in  diameter,  invariably  fixed  in  their  sty,  arid  the 

stars  passing  slowly  behind  it. 

The  nu>on  " But,"  says  Sir  John  Herschel,  "the  moon's  rotation  on 
revolves  .>n  j^  ^^  jg  unjform .  anj  since  her  motion  in  her  orbit  is  not 
so,  we  are  enabled  to  look  a  few  degrees  round  the  equatorial 
parts  of  her  visible  border,  on  the  eastern  or  western  side, 
according  to  circumstances ;  or,  in  other  words,  the  line  join- 
ing the  centers  of  the  earth  and  moon  fluctuates  a  little  in  its 
position,  from  its  mean  or  average  intersection  with  her  sur- 
face, to  the  east,  or  westward.  And,  moreover,  since  the 
axis  about  which  she  revolves  is  not  exactly  perpendicular  to 
her  orbit,  her  poles  come  alternately  into  view  for  a  small 
space  at  the  edges  of  her  disc.  These  phenomena  are  known 
by  the  name  of  libratiom.  In  consequence  of  these  two  dis- 
tinct kmds  of  libration,  the  same  identical  point  of  the  moon's 
surface  is  not  always  the  center  of  her  disc ;  and  we  therefore 
get  sight  of  a  zone  of  a  few  degrees  in  breadth  on  all  sides 
of  the  border,  beyond  an  exact  hemisphere." 


CHAPTER   III. 

THE  EARTH'S  ORBIT  ECCENTRIC.  —  THE  APPARENT  ANGULAR 

MOTION    OP   THE   SUN    NOT    UNIFORM. LAWS    BETWEEN    DIS- 
TANCE,     REAL,     AND     ANGULAR     MOTION. ECCENTRICITY     OP 

THE   ORBIT. 

CHAP,  in        (  69.  )  THE  gun's  parallax  is  too  small  to  be  detected  by 
The   sun  any  common  means  of  observation ;  hence  it  remained  un- 

tiufearth  *"  ^nowri'  ^or  a  'ong  series  of  years,  although  many  ingenious 
methods  were  proposed  to  discover  it.  The  only  decision 
that  ancient  astronomers  could  make  concerning  it  was,  that 
it  must  be  less  than  20"  or  15"  of  arc ;  for,  were  it  as  much 
as  that  quantity,  it  could  not  escape  observation. 

Now  let  us  suppose  that  the  sun's  horizontal  parallax  is  less 
than  20";  that  is,  the  apparent  semidiameter  of  the  earth,  as 
scpn  from  the  sun,  must  be  less  than  20";  but  the  semidia- 


APPARENT    DIAMETERS.  83 

meter  of  the  sun  is  15'  56",  or  956" ;  therefore  the  sun  must  CHAP.  m. 

be  vastly  larger  than  the  earth  —  by  at  least  48  times  its 

diame*ter ;  and  the  bulk  of  the  earth  must  be,  to  that  of  the 

sun,  in  as  high  a  ratio  as  1  to  the  cube  of  48.     But  as  we  do 

not  allow  ourselves  to  know  the  true  horizontal  parallax  of 

the  sun,  all  the  decision  we  can  make  on  this  subject  is,  that 

the  sun  is  vastly  larger  than  the  earth. 

(  70.  )  Previous  observations,  as  we  explained  in  the  first     Does   t.h«, 
section  of  this  work,  clearly  show,  or  give  the  appearance  of  ™*  elrtT^ 
the  sun  going  round  the  earth  once  in  a  year ;  but  the  appear-  the      eanh 
ance  would  be  the  same,  whether  the  earth  revolves  round  the  round      thc 
sun,  or  the  sun  round  the  earth,  or  both  bodies  revolve  round 
a  point  between  them.     We  are  now  to  consider  which  is  the 
most  probable :  (hat  a  large  body  should  circulate  round  a  much 
smaller  one;  or,  the  smaller  one  round  a  large  one.      The  last 
suggestion  corresponds  with  our  knowledge  and  experience  in 
mechanical  philosophy ;  the  first  is  opposed  to  it. 

(71.)  We  have  seen,  in  the  last  chapter,  that  the  semidia- 
meter  and  horizontal  parallax  of  a  body  have  a  constant  rela- 
tion to  each  other ;  and,  while  we  cannot  discover  the  one, 
we  will  examine  all  the  variations  of  the  other  ( if  it  have  va- 
riations ),  and  thereby  determine  whether  the  earth  and  sun 
always  remain  at  the  same  distance  from  each  other. 

Here  it  is  very  important  that  the  reader  should  clearly       Methods 
understand,  how  the  apparent  diameter  of  a  heavenly  body  of  measuring 

.  .   .  apparent  dia- 

can  be  determined  to  great  precision.  meters. 

As  an  example,  we  shall  take  the  diameter  of  the  sun ;  but 
the  same  principles  are  to  be  followed,  and  the  same  deduc- 
tions are  to  be  made,  whatever  body,  moon,  or  planet,  may  be 
under  observation. 

An  instrument  to  measure  the  apparent  diameter  of  a  planet  The  mtcio 
is  called  a  micrometer.  It  is  an  eyepiece  to  a  telescope,  with 
opening  and  closing  parallel  wires ;  the  amount  of  the  opening 
is  measured  by  a  mathematical  contrivance.  For  the  measure 
of  all  small  objects,  the  micrometer  is  exclusively  used;  and 
since  it  is  impossible  that  any  one  observation  can  be  relied 
upon  as  accurate  (  on  account  of  the  angular  space  eclipsed 
by  the  wires),  a  great  number  of  observations  are  taken,  and 


meter. 


84  ASTRONOMY. 

Clil?-  "_'•   the  mean  result  is  regarded  as  a  single  observation.     Gene- 
rally speaking,  the  following  method  is  more  to  be  relied  upon, 
when  large  angles  are  measured,  and  to  it  we  commend  special 
attention. 
The  me-      The  method  depends  on  the  time  employed  by  the  body  in  pass- 

thod  by  time  •      ^   perpendicular  wires  of  the  transit  instrument. 

IB       passing     v          r    r 

the  meridian.  All  bodies  (by  the  revolution  of  the  earth)  come  to  the 
meridian  at  right  angles,  and  15  degrees  pass  by  the  meridian 
in  one  hour  of  sidereal  time ;  and,  in  four  minutes,  one  de- 
gree will  pass;  and,  in  two  minutes  of  time,  30  minutes  of  arc 
will  pass  the  meridian  wire. 

Now  if  the  sun  is  on  the  equator,  and  stationary  there,  and 
employs  two  minutes  of  sidereal  time  in  passing  the  meridian, 
then  it  is  evident  that  its  apparent  diameter  is  just  30'  of  arc; 
if  the  time  is  more  than  two  minutes,  the  diameter  is  more; 
if  less,  less. 

But  we  have  just  made  a  supposition  that  is  not  true  ;  we 
have  supposed  the  sun  stationary,  in  respect  to  the  stars ;  but 
it  is  not  so :  it  apparently  moves  eastward ;  therefore  it  will 
not  get  past  the  meridian  wire  as  soon  as  it  would  if  station- 
ary. Hence  we  must  have  a  correction,  for  the  sun's  motion, 
applied  to  the  time  of  its  passing  the  meridian. 
Corrections  We  have  also  supposed  the  sun  on  the  equator,  and  for  a 

to  be  made.  moment  continue  the  supposition,  and  also  conceive  its  dia- 
meter to  be  just  30'  of  arc.  Now  suppose  it  brought  up  to 
the  20th  degree  of  declination,  on  that  parallel,  it  will  extend 
over  more  than  30'  of  arc,  because  meridians  converge  toward 
the  pole ;  therefore  the  farther  the  sun,  or  any  other  body  is  from 
tfie  equator,  the  longer  it  will  be  in  passing  the  meridian  on  thai1 
account;  the  increase  of  time  depending  on  the  cosine  of  tlie 
declination.  (See  59.) 

Hence  two  corrections  must  be  made  to  the  actual  time 
that  the  sun  occupies  in  crossing  the  meridian  wire,  before  we 
can  proportion  it  into  an  arc  :  one  for  the  progressive  motion 
of  the  sun  in  right  ascension ;  and  one  for  the  existing  decli- 
nation. We  give  an  example. 

Method  of        ^n    tlie    firSt    da^  °f  June'  1846'  ^6    Sidereal   time    (time 

the  measured  by  the  sidereal  clock  )  of  the  sun  passing  the  me- 


APPARENT    DIAMETERS.  rtft 

ridian  wire,  wa,s  observed  to  be  2  m.  16.64  s. ;  the  declination  CHAP.  HL 
was  22°  2'  45",  and  the  hourly  increase  of  right  ascension  was  exact   a(,pa 

10.235s.     What  was  the  sun's  semidiameter  ?  rent  diame- 

ter   of    the 

3600s.     :     10.235s.     ::     136.64    :     0.39  a.  sun,    mo«a 

or  planets. 

Observed  dura,  of  tran.,  in  sees.,  136.64 
Reduction  for  solar  motion,    -  .39 

136.25  .  .  log.  2.134337 
Dec.  22°  2'  45";  cosine,  9.967021 

Duration,  if  stationary  on  equa.,  126.3  s. .  .  log.  2.101358 

Minutes  or  seconds  of  time  can  be  changed  into  minutes  or 
seconds  of  arc,  by  multiplying  by  15 ;  therefore  the  diameter 
of  the  sun,  at  this  time,  subtended  an  arc  of  1894". 5,  and  its 
gemidiameter  947".2,  or  15'  47".2 ;  which  is  the  result  given 
in  the  Nautical  Almanac,  from  which  any  number  of  examples  of 
this  kind  can  be  taken.  We  give  one  more  example,  for  the 
benefit  of  those  who  may  not  have  a  Nautical  Almanac. 

On  the  30th  day  of  December  (  not  material  what  year ), 
the  sidereal  time  of  the  sun's  diameter  passing  the  meridian 
was  observed  to  be  2m.  22.2s.,  or  142.2s.  The  sun's 
hourly  motion  in  right  ascension,  at  that  time,  was  11.06  s., 
and  the  declination  was  23°  11'.  What  was  the  sun's  semi- 
diameter?*  Ans.  16'  17".3. 

These  observations  may  be  made  every  clear  day  through-       Extreme 

,  iiii  i  ,  i  raluesof  th« 

out  the  year ;  and  they  have  been  made  at  many  places,  and  gun,s    apl>a. 
for  many  years;  and  the  combined  results  show  that  the 

meter. 

*  The  following  is  the  formula  for  these  reductions  : 
15(i— c)cos.  D_ 
~R~ 

Here  t  is  the  observed  interval  in  seconds,  c  is  the  correction  for  the  in- 
crease in  right  ascension,  D  is  the  declination,  R  the  radius  of  the  tables, 
and  *  is  the  result  in  seconds  of  arc.  c  is  always  very  small ;  for  one 
hour,  or  3600  s.,  the  variation  is  never  less  than  8.976  s.,  nor  more  than 
11.11  s.  The  former  happens  about  the  middle  of  September  ;  the  lat- 
ter about  the  20th  of  December.  For  the  meridian  passage  of  the  moon, 
the  correction  c  is  considerable  ;  because  the  moon's  increase  of  right 
ascension  is  comparatively  very  rapid.  For  the  planets,  c  may  t*»  dii- 
n»garded. 


86  ASTRONOMY. 

CHAP,  in.  apparent  diameter  of  the  sun  is  the  same,  on  the  same  day  of 
the  year,  from  whatever  station  observed. 

The  least  semidiameter  is  15'  45".  1  ;  which  corresponds,  in 
time,  to  the  first  or  second  day  of  July  ;  and  the  greatest  is  16' 
17  ".3,  which  takes  place  on  the  1st  or  2d  of  January. 

Now  as  we  cannot  suppose  that  there  is  any  real  change  in 
the  diameter  of  the  sun,  we  must  impute  this  apparent  change 
to  real  change  in  the  distance  of  the  body,  as  explained  in 
Art.  G6. 

Variation      Therefore  the  distance  to  the  sun  on  the  30th  of  Decem- 

)e   dls"  ber,  must  be  to  its  distance  on  the  first  day  of  July,  as  the 

trie  earth  to  number  15'  45".  1  is  to  the  number  16'  17".3,  or  as  the  num- 

thesun.        ber  945  i  to  977.3  ;  and  all  other  days  in  the  year,  the  pro- 

portional  distance  must  be  represented  by  intermediate  num- 

bers. 

From  this,  we  perceive  that  the  sun  must  go  round  the 
earth,  or  the  earth  round  the  sun,  in  very  nearly  a  circle  ;  for 
were  a  representation  of  the  curve  drawn,  corresponding  to 
the  apparent  semidiameter  in  different  parts  of  the  orbit,  and 
placed  before  us,  the  eye  could  scarcely  detect  its  departure 
from  a  circle. 

(  72.)  It  should  be  observed  that  the  time  elapsed  between 
the  greatest  and  least  apparent  diameter  of  the  sun,  or  the 
reverse,  is  just  half  a  year  ;  and  the  change  in  the  sun's  lon- 
gitude is  180°. 

Eccentri-      If  we  would  consider  the  mean  distance  between  the  earth 
earth»sf0rwte  &n*^  sun  as  ww%  (as  *s  customary  with  astronomers),  and  then 
now  known,  put  x  to  represent  the  least  distance,  and  y  the  greatest  dis- 
tance, we  shall  have 


And,  -     -    x    :    y    :  :     9451     :     9773. 

A  solution  gives  *=0.98326,  nearly,  and  y=1.01674,  nearly; 
showing  that  the  leastjmean,  and  greatest  distance  to  the  sun, 
must  be  very  nearly  as  the  numbers  .98326,  1.,  and  1.01674. 
The  fractional  part,  .01674,  or  the  difference  between  the 
extremes  and  mean  (  when  the  ntean  is  unity  ),  is  called  th« 
eccentricity  of  the  orbit. 


SUN'S    MOTION    IN    LONGITUDE.  8r. 

The  tccentricity,  as  just  mentioned,  must  not  be  regarded  as  CHAP.  H 
accurate.     It  is  only  a  first  approximation,  deduced  from  the 
first  and  most  simple  view  of  the  subject;  but  we  shall,  here- 
after, give  other  expositions  that  will  lead  to  far  more  accu- 
rate results. 

In  theory,  the  apparent  diameters  are  sufficient  to  determine     Eccentric* 
the  eccentricity,  could  we  really  observe  them  to  rigorous  t^re^m  d"^ 
exactness ;  but  all  luminous  bodies  are  more  or  less  afl'ected  meters  only 
by  irradiation,  which  dilates  a  little  their  apparent  diameters ;  aPProximi 
and  the  exact  quantity  of  this  dilatation  is  not  yet  well 
ascertained. 

(  73. )  The  sun's  right  ascension  and  declination  can  be 
observed  from  any  observatory,  any  clear  day;  and  from 
thence  we  can  trace  its  path  along  the  celestial  concave  sphere 
above  us,  and  determine  its  change  from  day  to  day ;  and  we 
find  it  runs  along  a  great  circle  called  the  ecliptic,  which 
crosses  the  equator  at  opposite  points  in  the  heavens ;  and 
the  ecliptic  inclines  to  the  equator  with  an  angle  of  about 
23°  27'  40". 

The  plane  .of  the  ecliptic  passes  through  the  center  of  the 
earth,  showing  it  to  be  a  great  circle,  or,  what  is  the  same 
thing,  showing  that  the  apparent  motion  of  the  sun  has  its 
center  in  the  line  which  joins  the  earth  and  sun. 

The  apparent  motion  of  the  sun  along  the  ecliptic  is  called     Variations 
longitude ;  and  this  is  its  most  regular  motion.  m   the   dls" 

tance  of  the 

When  we  compare  the  sun  s  motion,  in  longitude,  with  its  8un>      com. 
semidiameter,  we  find  a  correspondence  —  at  least,  an  apparent  Pared    with 

its  variations 

connection.  in  iongitDde< 

When  the  semidiameter  is  greatest,  the  motion  in  longitude . 

is  greatest;  and,  when  the  semidiameter  is  least  the  motion 

in  longitude  is  least ;  but  the  two  variations  have  not  the  same 

ratio. 

When  the  sun  is  nearest  to  the  earth,  on  or  about  the  30th 

of  December,  it  changes  its  longitude,  in  a  mean  solar  day, 

1°  1'  9".95.     When  farthest  from  the  earth,  on  the  1st  of  . 

July,  its  change  of  longitude,  in  24  hours,  is  only  57'  11  ".48. 

A  uniform  motion,  for  the  whole  year,  is  found  to  be  59'  8".33. 
The  ancient  philosophers  contended  that  the  sun  moved 


88  ASTRONOMY. 

CHAP,  m.   about  the  earth  in  a  circular  orbit,  and  its  real  velocity  uni- 

form ;  Lut  the  earth  not  being  in  the  center  of  the  circle,  the 

same  portions  of  the  circle  would  appear  under  different  angles  ; 

and  hence  the  variation  in  its  apparent  angular  motion. 

The  result      NOW  jf  ^njs  jg  a  true  vjew  Of  tne  subiecf   the  variation  in 

»hows      that 

the  angular  angular  motion  must  be  in  exact  proportion  to  the  variation  in 
motion  is  in  distance,  as  explained  in  the  note  to  Art.  66;  that  is,  945".  1 
^portion"6  should  be  to  977".3,  as  57'  11".48  to  61'  9".95,  if  the  sup- 
to  the  square  position  of  the  first  observers  were  true.  But  these  numbers 
e  dls"  have  not  the  same  ratio  ;  therefore  this  supposition  is  not 
satisfactory  ;  and  it  was  probably  abandoned  for  the  want  of 
this  mathematical  support.  The  ratio  between  945".  1,  and 

0770 
977".3is  .....     -     -     -      ^=1.0341,  nearly: 


between  57'  11".48,  and6l'9".95,  f"        1.0694,  nearly. 


If  we  square  (1.0341)  the  first  ratio,  we  shall  have  1.06936, 
a  number  so  near  in  value  to  the  second  ratio,  that  we  con- 
clude it  ought  to  be  the  same,  and  would  be  the  same,  pro- 
vided we  had  perfect  accuracy  in  the  observations. 
Law  be.  Thus  we  compare  the  angular  motion  of  the  sun  in  difie- 
m]  d\a-  ren^  parts  of  its  orbit ;  and  we  always  find,  that  the  inverse 
square  of  its  distance  is  proportional  to  its  angular  motion;  and 
this  incontestable /ad  is  so  exact  and  so  regular,  that  we  lay 
it  down  as  a  law ;  and  if  solitary  observations  do  not  corre- 
spond with  it,  we  must  condemn  the  observations,  and  not 
the  law. 

( 74.)  To  investigate  this  su eject  thoroughly,  we  cannot 
avoid  making  use  of  a  little  geometry. 

Let  Fig.  17  represent  the  solar  orbit,*  the  sun  apparently 
revolving  about  the  observer  at  0.     The  distance  from  0  to 

*  We  say  solar  orbit,  when  it  is  really  the  earth's  orbit;  so  we  speak 
of  the  sun's  motion,  when  it  is  really  the  motion  of  the  earth  ;  and  it 
is  customary,  with  astronomers,  to  speak  of  apparent  motions  as  real 
and  none  object  to  this  manner  of  speaking,  who  have  a  clear  or  en- 
larged view  of  the  science — for  to  depart  from  it  would  lead  to  oft- 
repeated  and  troublesome  technicalities,  if  not  to  confusion  of  ideas 
Clearness  does  not  always  correspond  with  exactness  of  expression. 


VARIATIONS    IN    SOLAR    MOTION.  8S 

any  point  in  the  or- Fig.  17 CHAP.  in. 

bit  is  called  the  ra- 
dius vector;  audit  is| 
a  varying  quantity, 
conceived  to  sweep  I 
round  the  point  0. 

Let  D  be  the  va- 
lue of  tl^  radius  vec- 
tor at  any  point,  and 
rD  its  value  at  some 
other  point,  as  repre- 
sented in  the  figure.     Let  y  represent  the  real  motion  of  the     Variation! 
sun,  for  a  very  short  interval  of  time,  at  the  extremity  of  the  ™  "*' 
radius  vector  D\  and  x  represent  the  real  motion,  at  the  tion. 
extremity  of  the  radius  vector  r  D,  in  jthe  same  time. 

From  0,  as  a  center,  at  the  distance  of  unity,  describe  a 
circle.  Put  A  to  represent  the  angle  under  which  x  appears 
from  0;  then,  by  observation,  r2A  is  the  angle  under  which  y 
appears  from  the  same  point. 

Now,  considering  the  sectors  as  triangles,  we  have  the  fol- 
lowing proportions : 

1    :    A    ::    rD    :    0; 
:  r*A  :  :     D      :    y. 

From  the  first,     -     -     x=rAD, 

From  the  second,       -     y=r3AD. 

Multiply  the  first  of  these  equations  by  r,  and  we  perceive 
that     ------     y—rx. 

This  last  equation  shows  that  the  real  velocity  of  the  earth  The  real 
in  its  orbit  varies  in  the  inverse  ratio  as  the  radius  vector ;  or  J^0™*^,  °jn 
it  varies  directly  as  the  apparent  diameter  of  the  sun.  its  orbit  »». 

( 75.)  If  we  multiply  rl)  by  x,  the  product  will  express  the  j^,g  asa  ** 
double  of  an  area  passed  over  by  the  radius  vector  in  a  certain  rent   diam* 
interval  of  time ;  and  if  we  multiply  D  by  y,  we  shall  have  ter* 
the  double  of  another  area  passed  over  by  the  radius  vector  in 
the  same  time.     But  the  first  product  is  rDx,  and  the  second 
is  the  same,  as  we  shall  see  by  taking  the  value  of  y  (r  a?) ;  that 
is  rDx=rDx\  hence  we  announce  this  general  law: 


50  ASTRONOMY. 

CHAP,  in.        That  the  solar  radius  vector  describes  equal  areas  in  equal 

The  radius  times. 

•cribes  e  uli      When  expressed  in  more  general  terms,  this  is  one  of  the 
»reas   in  e-  three  laws  of  Kepler,  which  will  be  fully  brought  into  notice 

qnal  times.      JQ  ft  su|jsequent  parfc  of  this  WOrk. 

If  we  draw  lines  from  any  point  in  a  plane,  reciprocally 
proportional  to  the  sun's  apparent  diameter,  and  at  angles 
differing  as  the  change  of  the  sun's  longitude,  and  then  con- 
nect the  extremities  of  such  lines  made  all  round  the  point, 
the  connecting  lines  will  form  a  curve,  corresponding  with  an 
ellipse  (see  Fig.  18),  which  represents  the  apparent  solar  orbit ; 
and,  from  a  review  of  the  whole  subject,  we  give  the  follow- 
ing summary: 

Laws    of       1.   The  eccentricity  of  the  solar  ellipse,  as  determined  from  the 
^7  ^  an  aPParent  dwnwter  of  the  sur,  is  .01674.* 

2.  The  surfs  angular  velocity  varies  inversely  as  the  square 
of  its  distance  from  the  earth. 

3.  The  real  velocity  is  inversely  as  the  distance. 

4.  The  areas  described  by  the  radius  vector  are  proportional 
to  the  times  of  description. 

(76.)  We  have  several  times  mentioned,  that,  as  far  as 
appearances  are  concerned,  it  is  immaterial  whether  we  con- 
sider the  sun  moving  round  the  earth,  or  the  earth  round  the 
sun;  for,  if  the  earth  is  in  one  positipn  of  the  heavens,  the 

*,By  making  use  of  the  2d  principle,  above  cited,  we  can 
compute  the  eccentricity  of  the  orbit  to  greater  precision  than 
by  the  apparent  diameters,  because  the  samfr  error  of  obser- 
vation on  longitude  would  not  be  as  proportionally  great  as 
on  apparent  diameter. 

Let  E  be  the  eccentricity  of  the  orbit;  then  (1 — E)  is 
the  least  distance  to  the  sun,  and  (\-\-E)  the  greatest  dis- 
tance. Then,  by  observation,  we  have 

(\—Ey    :    (1+^)2    ::    57'  11".48    :    61'    9".95; 
Or,  (1— .ff)2    :    (1+.E)2     ::        343148      :       366995; 
Or,    l—E       :      1+E        :  :    V843148      :    -^366996; 

Whence  .#=.016788+.  We  shall  give  a  still  more  accu- 
rate method  of  computing  this  important  element. 


SUN'S    ELLIPTICAL    MOTION.  <)j 

Bun  appears  exactly  in  Fig-  18-  CHAP,  in. 

the  oppi  site  position, 

and  ever  y  motion 

made    by    the    earth 

must  correspond  to  an 

apparent  motion  made 

by  the  sun. 

But,  for  the  purpose 
of  getting   nearer  to 
fact,  we  will  now  sup- 
pose the  earth  revolves  round  the  sun  in  an  elliptical  orbit, 
as  represented  by  Fig.  18. 

We  have  very  much  exaggerated  the  eccentricity  of  the 
orbit,  for  the  purpose  of  bringing  principles  clearer  to  view. 

The  greatest  and  least  distances,  from  the  sun  to  the  earth, 
make  a  straight  line  through  the  sun,  and  cut  the  orbit  into 
two  equal  parts.  When  the  earth  is  at  B,  the  greatest  dis- 
tance from  the  sun,  it  is  said  to  be  in  apogee,  and  when  at  A, 
the  least  distance,  it  is  in  perigee ;  and  the  line  joining  the 
apogee  and  perigee  is  the  major,  or  greater  diameter  of  the 
orbit ;  arid  it  is  the  only  diameter  passing  through  the  sun,  that 
cuts  the  orbit  into  two  equal  parts. 

Now,  as  equal  areas  are  described  in  equal  times,  it  follows  Observa- 
that  the  earth  must  be  just  half  a  year  in  passing  from  apogee  JJJ  *  ^" 
to  perigee,  and  from  perigee  to  apogee ;  provided  that  these  positions  of 
points  are  stationary  in  the  heavens,  and  they  are  so,  very  the  8olar  a> 

*   pogee      and 
nearly.*  perigee. 

If  we  suppose  the  earth  moves  along  the  orbit  from  D  to 
A,  and  we  observe  the  sun  from  D,  and  continue  observa- 
tions upon  it  until  the  earth  comes  to  C,  then  the  longitude 
of  the  sun  has  changed  180°;  and  if  the  time  is  less  than 


*  The  longer  axis  of  the  orbit,  or  apogee  point,  changes  position  by 
b.  very  slow  motion  of  about  12"  per  annum,  to  the  eastward  :  but  this 
motion  must  be  disregarded,  for  the  present,  as  well  as  many  other  mi- 
nute deviations,  to  be  brought  into  view  when  we  are  better  prepared 
to  understand  them. 

Those  minute  variations/ for  short  periods  of  time,  do  not  sensibly 
affect  general  results. 


92  ASTRONOMY. 

CHA».  HI.  half  a  year,  we  are  sure  the  perigee  is  in  this  part  of  the 
orbit.  If  we  continue  observations  round  and  round,  and 
find  where  180  degrees  of  longitude  correspond  with  half  a 
year,  there  will  be  the  position  of  the  longer  axis ;  which  is 
sometimes  called  the  line  of  the  apsides. 

Difficulties,  \ye  cannot  determine  the  exact  point  of  the  apogee  or 
perigee,  by  direct  observations  on  the  sun's  apparent  diame- 
ter; for  about  these  points  the  variations  are  extremely  slow 
and  imperceptible. 

If  we  take  observations  in  respect  to  the  sun's  longitude, 
when  the  earth  is  at  b,  and  watch  for  the  opposite  longitude, 
when  the  earth  is  about  a,  and  find  that  the  area  b  Da  was 
described  in  little  less  thafti  half  a  year,  and  the  area  aCb,  in 
a  little  more  than  half  a  year,  then  we  know  that  b  is  very 
near  the  apogee,  and  a  very  near  the  perigee. 

If  we  take  another  point,  b',  and  its  opposite,  a',  and  find 
converse  results,  then  we  know  that  the  apogee  is  between 
the  points  b'  and  b,  and  we  can  proportion  to  it  to  great  exact- 
ness. 
Longitude      (  77. )  The  longitude  of  the  apogee,  for  the  year  1801,  was 

Md  pe'rfcfT  "°  31/  9"'  and>  of  course»  tne  perigee  was  in  longitude  279° 
31'  9".  These  points  move  forward,  in  respect  to  the  stars, 
about  12"  annually,  and,  in  respect  to  the  equinox,  about  62" ; 
more  exactly  61". 905,  and,  of  course,  this  is  their  annual 
increase  of  longitude. 

In  the  year  1250,  the  perigee  of  the  sun  coincided  with  the 
winter  solstice,  and  the  apogee  with  the  summer  solstice ;  and 
at  that  time  the  sun  was  178  days  and  about  17^  hours  on 
the  south  side  of  the  equator,  and  186  days  and  about  12^ 
hours  on  the  north  side ;  being  longer  in  the  northern  hemi- 
sphere than  in  the  southern,  by  seven  days  and  19  hours:  at 
present,  the  excess  is  seven  days  and  near  17  hours. 
The  year  (78.)  As  the  sun  is  a  longer  time  in  the  northern  than  in 
the  southern  hemisphere,  the  first  impression  might  be,  that 
more  solar  heat  is  received  in  one  hemisphere  than  in  the 
other ;  but  the  amount  is  the  same ;  for  whatever  is  gained 
in  time,  is  lost  in  distance ;  and  what  is  lost  in  time,  is  gained 
by  a  decrease  of  distance.  The  amount  of  heat  depend1}  OB 


SUN'S    ELLIPTICAL    MOTION.  y^ 

the  intensity  multiplied  by  the  time  it  is  applied;  and  the  CHAP.  ra. 
product  of  the  time  .and  distanced  the  sun,  is  the  same  in 
either  hemisphere  ;  but  the  amount  of  heat  received,  for  a 
single  day,  is  different  in  the  two  hemispheres. 

(  79.)  Conceive  a  line  drawn  through  the  sun,  at  right 
angles  to  the  greater  diameter  of  the  orbit  D  S  C  (  see  Fig. 
18  ),  the  point  C  is  8°  21'  from  the  first  point  of  Aries;  and 
if  we  observe  the  time  occupied  by  the  sun  in  describing  180 
degrees  of  longitude,  from  this  point  (or  from  any  point  very 
near  this  point),  that  time,  taken  from  the  whole  year,  will 
give  the  time  of  describing  the  other  180  degrees. 

Without  being  very  minute,  we  venture  to  state,  that  the     A  method 
time  of  describing  the  arc  DA  C  is  178  days  17^  hours;  and  of  obtaimn« 
the  time  of  describing  the  arc   CBD  is  186  days  12-1  hours,  city  of  an  or- 
But,  as  areas  are  in  proportion  to  the  times  of  their  descrip-  bit* 
tion  ;  therefore, 

d.        h.  d.        h. 

area  CSDA  :  area  CBDS  :  :  178  17£  :  186  12i. 

By  taking  half  of  the  greater  axis  of  the  ellipse  equal 
unity,  and  the  eccentricity  an  unknown  quantity,  e,  the 
mathematician  can  soon  obtain  analytical  expressions  for 
the  two  areas  in  question;  and  then,  from  the  proportion, 
he  can  find  the  value  of  the  eccentricity  e  :  but  there  is  a 
better  method  —  we  only  give  an  outside  view  of  this,  for  the 
light  it  throws  on  the  general  principle. 

(  80.)  Now  let  us  conceive  the  orbit  of  the  earth  inclosed 
by  a  circle  whose  diameter  is  the  greatest  dkmeter  of  the 
ellipse,  as  represented  by  Fig.  19. 

For  the  sake  of  simplicity  we  will  suppose  the  observer  at 


rest  at  the  point  o  (  one  focus  of  the  ellipse  ),  and  the  sun  *ion  for  fimi> 
really  to  move  round  on  the  ellipse,  describing  equal  areas  Ration  "i! 
in  equal  times  round  the  point  o.  an  e">p»«- 

Conceive,  also,  an  imaginary  sun  to  pass  round  the  circle, 
describing  equal  angles,  in  equal  times,  round  the  center  n/i. 
Now  suppose  the  two  suns  to  be  together  at  the  point  E  :  — 
they  depart,  one  on  the  ellipse,  the  other  on  the  circle;  and, 
on  account  of  both  describing  equal  areas,  in  equal  times, 
round  their  respective  centers  of  motion,  they  will  be  together 


94 


ASTRONOMY. 


CH 


Fig.  19. 


at  the  point  A,  ami 
again  at  the  point  B} 
and  so  continue  in 
each  subsequent  re- 
volution. 

The  imaginary  sun 
on  the  circle  every- 
where describes  equal 
angles  in  equal  times ; 
and  the  true  sun,  on 
the  ellipse,  describes 
only  equal  areas  in 
equal  times ;  but  the  angles  will  be  unequal.  Conceive  the 
two  suns  to  depart,  at  the  same  time,  from  the  point  B, 
and,  after  a  certain  interval  of  time,  one  is  at  s,  the  other  at 
s'.  Then  we  must  have 

area  oBs  :  area  mBs'  :  :  area  ellipse  :  area  circle. 
Mean  and  The  angle  Bms'  is  the  angle  the  sun  would  make,  or  its 
increase  in  longitude  from  the  apogee ;  provided  the  angular 
motion  of  the  sun  was  uniform.  The  angle  Bos  is  its  .true 
increase  01  longitude ;  the  difference  between  these  two  angles 
is  the  angle  in  n  o. 

The  angle  Bms'  is  always  known  by  the  time;  and  if  to 
every  degree  of  the  angle  Bms'  we  knew  the  corresponding 
angle  mno,  the  two  would  give  us  the  angle  Bos;  for, 

Bms' — mno=mon,  or    Bos. 

The  angle  Bms'  is  called  the  mean  anomaly,  and  the  angle 
B  o  s  is  called  the  true  anomaly. 

The  equa-      The  angle  Bms'  is  greater  than  the  angle  Bos,  all  the 
n  of  the  way  from  foQ  apogee  to  the  perigee;  but  from  the  perigee  to 
the  apogee, the  true  sun,  on  the  ellipse,  is  in  advance  of  the 
imaginary  sun  on  the  circle. 

The  angle  m  n  o  is  called  the  equation  of  the  center  ;  that  is, 
it  is  the  angle  to  be  applied  to  the  angle  about  the  center  m, 
to  make  it  equal  to  the  true  anomaly. 

The  angle  mno  depends  on  the  eccentricity  of  the  ellipse; 
and  its  amount  is  put  in  a  table  corresponding  to  every 


true 
•air 


ECCENTRICITY    OF    ORBIT.  95 

degree  of  the  mean  anomaly ;  suhtractive,  from  the  apogee  to  CHJU-.  ni 
the  perigee,  and  additive  from  the  perigee  to  the  apogee.* 

(81.)  Again:  conceive  the  two  suns  to  set  out  from  the  same     The  great, 
point.  B:  and  as  the  angle  Bms'  increases  uniformly,  it  will  est  eiuatlon 

J\  ofthecentei 

increase  and  become  greater  and  greater  than  the  angle  h  o  s,  sives  the  ec- 
until  the  true  sun  attains  its  mean  angular  motion,  and  no  centncity  of 

mi  i  i  •  i 

longer.  I  hen  the  angle  m  n  o  attains  its  greatest  value,  and, 
at  that  time  the  side  mn=no,  and  the  point  n  is  over  the 
center  of  o  m,  and  o  s'  is  a  mean  proportional  between  o  B 
ind  o  A.  That  is,  when  the  sun,  or  any  planet,  attains  the 
greatest  equation  of  the  center,  the  true  sun  is  very  near  the 
txtremity  of  the  shorter  axis  of  the  ellipse  :  o,  the  greatest 
equation  of  the  center,  can  be  determined  by  observation ; 
and,  from  the  greatest  equation,  we  have  the  most  accurate 
method  of  computing  the  eccentricity  of  the  ellipse,  as  we 
may  see  by  the  note  below. f 

t  Let  C  (Fig.  20)  be  the 
place  of  the  true  sun,  and  G\ 
the  place  of  the  imaginary! 
sun  ;  the  line  m  F  cuts  off 
equal  portions  of  the  circle 
and  the  ellipse.     Then  we 
have    to    make   the    sector] 
m  F  G  to  the  triangle  om  C;  ^^^^^^^ 
as  the  circle  is  to  the  ellipse.     Now  let 

mB—a,     mC=b,     vm=ea,     5r=3il416; 
Then,  the  area  of  the  circle  is  *a? ;  the  area  of  the  ellipse  is 
*ab ;  that  of  the  sector  is  (  OF)^.  and  of  the  triangle  ^-. 

r  Hence,-     5?    :    O 


*  By  a  mere  mechanical  contrivance,  the  modern  astronomical  tables 
are  so  arranged,  that  all  corrections  are  rendered  additive  ;  so  that  the 
mechanical  oporator  cannot  make  a  mistake,  as  to  signs,  and  he  may 
continue  to  work  without  stopping  to  think.  These  arrangements 
have  their  advantages,  but  they  cover  up  and  obscure  principles. 


96  A  S  T  R  O  N  0  M  Y. 

CHAP.  ni.  When  once  the  eccentricity  of  any  planetary  ellipse  be- 
comes known,  the  equation  of  the  center,  corresponding  to  all 
degrees  of  the  mean  anomaly,  can  be  computed  and  put  into 
a  table  for  future  use ;  but  this  labor  of  constructing  tables 
belongs  exclusively  to  the  mathematician. 


Method  of      Or,-     -     eab    :      (GF)a    ::     b     :     a; 

deducing  the 

eccentricity  Or,  CO.       I  GF  II        1       I       1  . 

greatest     e-  Consequently,  GF=ea,  and  FG=om  ;  which  shows  that  the 
quation     of  angle  o  Cm  is  nearly  equal  to  Fm  G,  unless  it  is  a  very  eccen- 
tric ellipse.     Now  we  must  compute  the  number  of  degrees 
in  the  arc  FG.     tfhe  whole  circumference  is  %7ra. 
Therefore,  2»a     :     ea     :  :     360     :     arcFG: 

Hence,    -     -     -     arc  FG=  --  =angle  nm  C. 

rr 

But  the  angle   onm=nm  C-\-n  Cm=2nm  C,  nearly; 

Therefore,  —     —  =2  nmC=onm=  greatest  equation  of 

center,  nearly. 

But  the  greatest  equation  of  the  center,  for  the  solar  orbit, 
is,  by  observation,  1°  55'  30"  ;  and  as  the  sun  has  not  quite 
its  greatest  equation  of  the  center,  when  at  the  point  C,  it  will 
be  more  accurate  to  put 

=l°  55'  24". 


From  this  equation,  it  is  true,  we  have  only  the  approxi- 
mate value  of  e  ;  but  it  is  a  very  approximate  value,  and  suffi- 
ciently accurate. 

Reducing  both  members  to  seconds,  fand  we  have, 
3600-360  *=6924?r,     and    e  =0.0167842. 

The  greatest  equation  of  the  center  is  at  present  diminish 
ing  at  the  rate  of  17".  17  in  one  hundred  years:  this  corre- 
sponds to  a  diminution  of  eccentricity  by  0.00004166.  whir). 
is  determined  by  a  solution  of  the  following  equation  : 


CHANGE    OF    SEASONS. 


97 


CHAPTER    IV. 


THE    CAUSES    OF    THE    CHANGE   OF    SEASONS. 


(  82.  )  THE  annual  revolution  of  the  earth  in  its  orbit,  CHA*.  iv 
combined  with  the  position  of  the  earth's  axis  to  the  plane 
of  its  orbit,  produces  the  change  of  the  seasons. 

If  the  axis  were  perpendicular  to  the  plane  of  its  orbit,     The  cause 
there  would  be  no  change  of  seasons,  and  the  sun  would  then  °^thechan«t 

of  season*. 

be  all  the  while  in  the  celestial  equator. 

This  will  be  understood  by  Fig.  21.     Conceive  the  plane 
of  the  paper  to  be  the  plane  of  the  earth's  orbit,  and  conceive 
the  several  representations  of  the  earth's  axis,  JV/S',  to'  be  in- 
clined to  the  paper  at  an  angle  of  66°  32'. 
Fig.  21. 


In  all  representations  of  NS,  one  half  of  it  is  supposed  to 
bo  above  the  paper,  the  other  half  below  it. 

yS  is  always  parallel  to  itself;  that  is,  it  is  always  in  the 
same  position*  —  always  at  the  same  inclination  to  the  plane 

•  Except  minute  variations,  which  it  would  be  improper  to  notice  in 
this  part  of  the  work. 

7 


08  ASTRONOMY. 

CHAP,  iv.   Of  its  orbit  —  always  directed  to  the  same  point  in  the  hea- 
vens, in  whatever  part  of  the  orbit  it  may  be. 

The  plane  of  the  equator,  represented  by  Eq,  is  inclined  to 
the  plane  of  the  orbit  by  an  angle  of  23°  28'. 

inspecting  the  figure,  the  reader  will  gather  a  clearer 
°f  ^e  subject  than  by  whole  pages  of  description :  he 
will  perceive  the  reason  why  the  sun  must  shine  over  the 
north  pole,  in  one  part  of  its  orbit,  and  fall  as  far  short  of 
that  point  when  in  the  opposite  part  of  its  orbit ;  and  the 
number  of  degrees  of  this  variation  depends,  of  course,  on  the 
position  of  the  axis  to  the  plane  of  the  orbit. 

Positioner     Now  conceive  the  line  N S  to  stand  perpendicular  to  the 

to  plane  of  the  paper,  and  continue  so ;  then  Eq  would  lie  on 

change     of  the  paper,  and  the  sun  would  at  all  times  be  in  the  plane  of 

••aions.       the  equator,  and  there  would  be  no  change  of  seasons.     If 

N  S  were  more  inclined  from  the  perpendicular  than  it  now 

is,  then  we  should  have  a  greater  change  of  seasons. 

By  inspecting  the  figure,  we  perceive,  also,  that  when  it  is 
summer  in  the  northern  hemisphere,  it  is  winter  in  the 
southern ;  and  conversely,  when  it  is  winter  in  the  northern, 
it  is  summer  in  the  southern. 

When  a  line  from  the  sun  makes  a  right  angle  with  the 
earth's  axis,  as  it  must  do  in  two  opposite  points  of  its  orbit, 
the  sun  will  shine  equally  on  both  poles ,  and  it  is  then  in  the 
plane  of  the  equator ;  which  gives  equal  day  and  night  the 
world  over. 

Equal  days  and  nights,  for  all  places,  happen  on  the  20th 
of  March  of  each  year,  and  on  the  22d  or  23d  of  September. 
At  these  times  the  sun  crosses  the  celestial  equator,  and  is 
said  to  be  in  the  equinox. 

The  equi.      The  longitude  of  the  sun,  at  the  vernal  equinox,  is  0° ;  and 
•octiai    and  at  t]ie  autumnal  equinox,  its  longitude  is  180°. 
point"  The  time  of  the  greatest  north  declination  is  the  20th  of 

June ;  the  sun's  longitude  is  then  90°,  and  is  said  to  be  at 
the  summer  solstice. 

The  time  of  the  greatest  south  declination  is  the  22d  of 
December ;  the  sun's  longitude,  at  that  time,  is  270°,  and 
is  said  to  be  at  the  winter  solstice. 


CHANGES    OF   SEASONS.  99 

By  inspecting  the  figure,  we  perceive,  that  when  the  earth   CHAP.  iv. 
is  at  the  summer  solstice,  the  north  pole,  N,  and  a  conside-     LCD*  se» 
rable  portion  of  the  earth's  surface  around,  is  within  the  en-  i0ns  of  snn* 
lightened  half  of  the  earth ;  and  as  the  earth  revolves  on  its  aLkness  "It 
axis  JV/S',  this  portion  constantly  remains  enlightened,  giving  and     abou 
a  constant  day  —  or  a  day  of  weeks  and  months  duration, l  e  po  e8' 
according  as  any  particular  point  is  nearer  or  more  remote 
from  the  pole:  the  pole  itself  is  enlightened  full  six  months 
in  the  year,  and  the  circle  of  more  than  24  hours  constant 
sunlight  extends  to  23°  28'  from  the  pole  (not  estimating  the 
effects  of  refraction).     On  the  other  hand,  the  opposite,  or 
south  pole,  S,  is  in  a  long  season  of  darkness,  from  which  it 
can  be  relieved  only  by  the  earth  changing  position  in  its 
orbit. 

"  Now,  the  temperature  of  any  par$  of  the  earth's  surface 
depends  mainly,  if  not  entirely,  on  its  exposure  to  the  sun's  earth. 
rays.  Whenever  the  sun  is  above  the  horizon  of  any  place, 
that  place  is  receiving  heat ;  when  below,  parting  with  it,  by. 
the  process  called  radiation;  and  the  whole  quantities  re- 
ceived and  parted  with  in  the  year  must  balance  each  other 
at  every  station,  or  the  equilibrium  of  temperature  would  not 
be  supported.  Whenever,  then,  the  sun  remains  more  than 
12  hours  above  the  horizon  of  any  place,  and  less  beneath, 
the  general  temperature  of  that  place  will  be  above  the  ave- 
rage ;  when  the  reverse,  below.  As  the  earth,  then,  moves 
from  A  to  B,  the  days  growing  longer,  and  the  nights  shorter 
in  the  northern  hemisphere,  the  temperature  of  every  part  of 
that  hemisphere  increases,  and  we  pass  from  spring  to  sum- 
mer, while  at  the  same  time  the  reverse  obtains  in  the  southern 
hemisphere.  As  the  earth  passes  from  B  to  C,  the  days  and 
nights  again  approach  to  equality — the  excess  of  temperature 
in  the  northern  hemisphere,  above  the  mean  state,  grows  less, 
as  well  as  its  defect  in  the  southern ;  and  at  the  autumnal 
equinox,  C,  the  mean  state  is  once  more  attained.  From 
thence  to  D,  and,  finally,  round  again  to  A,  all  the  same  phe- 
nomena, it  is  obvious,  must  again  occur,  but  reversed; it  being 
now  winter  in  the  northern,  and  summer  in  the  southern 
hemisphere." 


100  ASTKONOMY. 

CHAP.  rv.  The  inquiry  is  sometimes  made  why  we  do  not  have  the 
warmest  weather  about  the  summer  solstice,  and  the  coldest 
weather  about  the  time  of  the  winter  solstice. 

Times  of  This  would  be  the  case  if  the  sun  immediately  ceased  to 
tem  give  extra  warmth,  on  arriving  at  the  summer  solstice ;  but 
if  it  could  radiate  extra  heat  to  warm  the  earth  three  weeks, 
before  it  came  to  the  solstice,  it  would  give  the  same  extra 
heat  three  weeks  after ;  and  the  northern  portion  of  the  earth 
must  continue  to  increase  in  temperature  as  long  as  the  sun 
continues  to  radiate  more  than  its  medium  degree  of  heat 
over  the  surface,  at  any  particular  place.  Conversely,  the 
whole  region  of  country  continues  to  grow  cold  as  long  as 
the  sun  radiates  less  than  its  mean  annual  degree  of  heat 
over  that  region.  The  medium  degree  of  heat,  for  the  whole 
year,  and  for  all  places,  of  course,  takes  place  when  the  sun 
is  on  the  equator;  the  average  temperature,  at  the  time  of 
the  two  equinoxes.  The  medium  degree  of  heat,  for  our 
.northern  summer,  considering  only  two  seasons  in  the  year, 
takes  place  when  the  sun's  declination  is  about  12  degrees 
north ;  and  the  medium  degree  of  heat,  for  winter,  takes  place 
when  the  sun's  declination  is  about  12  degrees  south ;  and 
if  this  be  true,  the  heat  of  summer  will  begin  to  decrease 
about  the  20th  of  August,  and  the  cold  of  winter  must  essen- 
tially abate  on  or  about  the  16th  of  February,  in  all  northern 
latitudes. 


CHAPTER   V. 

EQUATION     OF     TIME. 

(  83.)  WE  now  come  to  one  of  the  most  important  subjects 
in  astronomy — the  equation  of  time. 

Without  a  good  knowledge  of  this  subject,  there  will  be 
constant  confusion  in  the  minds  of  the  pupils ;  and  such  is 
the  nature  of  the  case,  that  it  is  difficult  to  understand  even 
the/wfe,  without  investigating  their  causes. 

Bid»r«ai      Sidereal  time  has  no  equation  ;  it  is  uniform,  and,  of  itself 
time  iwrftct  perfect  and  complete. 


EQUATION    OF    TIME.  101 

The  time,  by  a  perfect  clock,  is  theoretically  perfect  and  CHAI.  iv. 
complete,  and  is  called  mean  time. 

The  time,  by  the  sun,  is  not  uniform;  and,  to  make  it    Solar  time 
agree  with  the  perfect  clock,  requires  a  correction  —  a  quan-  notumform 
tity  to  make  equality;  and  this  quantity  is  called  the  equa- 
tion of  time.* 

If  the  sun  were  stationary  in  the  heavens,  like  a  star,  it 
would  come  to  the  meridian  after  exact  and  equal  intervals 
of  time;  and,  in  that  case,  there  would  be  no  equation  of 
time. 

If  the  sun's  motion,  in  right  ascension,  were  uniform,  then 
it  would  also  come  to  the  meridian  after  equal  intervals  of 
time,  and  there  would  still  be  no  equation  of  time.  But 
( speaking  in  relation  to  appearances  )  the  sun  is  not  station- 
ary in  the  heavens,  nor  does  it  move  uniformly ;  therefore  it 
cannot  come  to  the  meridian  at  equal  intervals  of  time,  and, 
of  course,  the  solar  days  must  be  slightly  unequal. 

When  the  sun  is  on  the  meridian,  it  is  then  apparent  noon,     Mean  sod 
for  that  day :  it  is  the  real  solar  noon,  or,  as  near  as  may  be,  *PPar5nt 
half  way  between  sunrise  and  sunset ;    but  it  may  not  be 
noon  by  the  perfect  clock,  which  runs  hypothetically  true  and 
uniform  throughout  the  whole  year. 

A  fixed  star  comes  to  the  meridian  at  the  expiration  of 
uvery  23  h.  56  m.  04.09  s.  of  mean  solar  time  ;  and  if  the  sun 
were  stationary  in  the  heavens,  it  would  come  to  the  meridian 
after  every  expiration  of  just  that  same  interval.  But  the 
sun  increaws  its  right  ascension  every  day,  by  its  apparent 
eastward  motion ;  and  this  increases  the  time  of  its  coming 
to  the  meridian ;  and  the  mean  interval  between  its  successive 
transits  o^er  the  meridian  is  just  24  hours ;  but  the  actual 
intervals'  are  variable  —  some  less,  and  some  more  than  24 
hours. 

On  and  about  the  1st  of  April,  the  time  from  one  meridian 
of  the  win  to  another,  as  measured  by  a  perfect  clock,  is  23  h. 
59  m  52.4  s. ;  less  than  24  hours  by  about  8  seconds.  Here, 
theo,  the  sun  and  clock  must  be  constantly  separating.  On 

•  In  astronomy,  the  term  equation  is  applied  to  all  corrections  to 
convert  a  mean  to  its  true  quantity. 


102  ASTRONOMY. 

CHAP.  v.    and  about  the  20th  of  December,  the  time  from  one  meridian 

of  the  sun  to  another  is  24  h.  Om.  24.3s.,  more  than  24 

seconds  over  24  hours ;  and  this,  in  a  few  days,  increases  to 

minutes — and  thus  we  perceive  the  fact  of  equation  of  time. 

Equation      To  detect  the  law  of  this  variation,  and  find  its  amount, 

of  time  the  we  must/  separate  the  cause  into  its  two  natural  divisions. 

result  of  two 

causes.  1.   The  unequal  apparent  motion  of  the  sun  along  the  ecliptic. 

2.   The  variable  inclination  of  this  motion  to  the  equator. 

If  the  sun's  apparent  motion  along  the  ecliptic  were  uni- 
form, still  there  would  be  an  equation  of  time ;  for  that  mo- 
tion, in  some  parts  of  the  orbit,  is  oblique  to  the  equator,  and, 
in  other  parts,  parallel  with  it;  and  its  eastward  motion,  in 
right  ascension,  would  be  greatest  when  moving  parallel  with 
the  equator. 

From  the  first  cause,  separately  considered,  the  sun  and 
clock  would  agree  two  days  in  a  year —  the  1st  of  July  and 
the  30th  of  December. 

From  the  second  cause,  separately  considered,  the  sun  and 
clock  agree  four  days  in  a  year — the  days  when  the  sun 
crosses  the  equator,  and  the  days  he  reaches  the  solstitial 
points. 

When  the  results  of  these  two  causes  are  combined,  the 
sun  and  clock  will  agree  four  days  in  the  year ;  but  it  is  on 
neither  of  those  days  marked  out  by  the  separate  causes ;  and 
the  intervals  between  the  several  periods,  and  the  amount  of 
the  equation,  appear   to  want  regularity  and  symmetry. 
Days  in      The  four  days  in  the  year  on  which  the  sun  and  clock 
1116  h^thl*  a<=ree'  *na*  *s'  snow  noon  at  the  same  instant,  are  April  15th, 
sun        and  June  16th,  September  1st,  and  December  24th. 

The  greatest  amount,  arising  from  the  first  cause,  is  7m. 
42s.,  and  the  greatest  amount,  from  the  second  cause,  is  9m. 
53  s. ;  but  as  these  maximum  results  never  happen  exactly  at 
the  same  time,  therefore  the  equation  of  time  can  never 
amount  to  17m.  35s.  In  fact,  the  greatest  amount  is  16m. 
17  s.,  and  takes  place  on  the  3d  of  November  ;  and,  for  a  long 
time  to  come,  the  maximum  value  will  take  place  on  the  same 
day  of  each  year ;  but,  in  the  course  of  ages,  it  will  vary  in 
its  amount  and  in  the  time  of  the  year  in  which  the  sun  and 


EQUATION    OF    TIME.  10 

clock  agree,  in  consequence  of  the  slow  and  gradual  change    CHAP.  ?. 
in  the  position  of  the  solar  apogee.     (See  Art.  77.) 

(  84.  )  The  elliptical  form  of  the  earth's  orbit  gives  rise  to    The 


cause. 


the  unequal  motion  of  the  earth  in  its  orbit,  and  thence  to  the  tlon  of  th* 

«un'«  center, 

apparent  unequal  motion  of  the  sun  in  the  ecliptic;  and  this  and  the  firm 
same  unequal  motion  is  what  we  have  denominated  the  first  Part  of  tho 
cause  of  the  equation  of  time.  Indeed,  this  part  of  the  equa-  J^1  "have 
tion  of  time  is  nothing  more  than  the  equation  of  the  center  a  common 
(  80),  changed  into  time  at  the  rate  of  four  minutes  to  a  degree. 
The  greatest  equation  for  the  sun's  longitude  (81,  note  ), 
is  by  observation  1°  55'  30";  and  this,  proportioned  into 
time,  gives  7  m.  42s.,  for  the  maximum  effect  in  the  equation 
of  time  arising  from  the  sun's  unequal  motion.  When  the  sun 
departs  from  its  perigee,  its  motion  is  greater  than  the  mean 
rate,  and,  of  course,  comes  to  the  meridian  later  than  it  other- 
wise would.  In  such  cases,  the  sun  is  said  to  be  slow  —  and 
it  is  slow  all  the  way  from  its  perigee  to  its  apogee  ;  and  fast 
in  the  other  half  of  its  orbit 

For  a  more  particular  explanation  of  the  second  cause,  we 

must  call  attention  to  Fig.  22 

Let  V  <£  ^   (Fig.  Fig  22. 

22  )    represent    the  ^ 

ecliptic,  and 

the  equator. 

By  the  first  cor- 

rection, the  apparent  j 

motion  along   the 

ecliptic    is    rendered 

uniform  ;  and  the  sun 

is   then  supposed  to 

pass  over  equal  spaces 

in  equal  intervals  of 

time   along   the    arc 

qp  £55.     But  equal 

spaces  of  arc,  on  the  ecliptic,  do  not  correspond  with  equal 

spaces  on  the  equator.     In  short,  the  points  on  the  ecliptic 

must  be  reduced   to  corresponding  points  on  the  equator. 

For  instance,  the  number  of  degrees  represented  by  T  £  on 


104  ASTRONOMY. 


,  v.  the  ecliptic,  is  greater  than  to  the  same  meridian  along  the 
equator.  The  difference  between  <ip£and  <¥>$>',  turned  into 
time,  is  the  equation  of  time  arising  from  the  obliquity  of  the 
ecliptic  corresponding  to  the  point  S. 

At  the  points  qp,  2S,  and  =o=,  and  also  at  the  southern 
tropic,  the  ecliptic  and  the  equator  correspond  to  the  same 
meridian  ;  but  all  other  equal  distances,  on  the  ecliptic  and 
equator,  are  included  by  different  meridians. 

HOW   t«      To  compute  the  equation  of  time  arising  from  this  cause, 

compute  the  we  must  solve  the  spherical  triangle  <¥>S  S'  ;    9P  /Sis  the  sun  'a 

of  the  eqna-  longitude,  and  the  angle  at  T  is  the  obliquity  of  thfc  elliptic, 

Uonoftim«.  and  at  S'  is  a  right  angle.     Assume  any  longitude,  a?  32°, 

35°,  or  40°,  or  any  other  number  of  degrees,  and  compute 

the  base.     The  difference  between  this  base  and  the  pun's 

longitude,  converted  into  time,  is  the  quantity  sought  corre- 

sponding to  the  assumed  longitude  ;  and  by  assuming  every 

degree  in  the  first  quadrant,  and  putting  the  result  in  a  table, 

we  have  the  amount  for  every  degree  of  the  entire  circle,  for 

all  the  quadrants  are  symmetrical,  and  the  same  distance  from 

either  equinox  will  be  the  same  amount. 

What  is  The  perfect  clock,  or  mean  time,  corresponds  with  the 
meant  by  sun  equator;  and  as  uniform  spaces  along  the  equator,  near  the 
•f  clock.  point  T,  will  pass  over  more  meridians  than  the  same  num- 
ber of  equal  spaces  on  the  ecliptic  ;  therefore  the  sun,  at  S, 
will  be  fast  of  clock,  or  come  to  the  meridian  before  it  ia  noon 
by  the  clock  —  and  this  will  be  true  all  the  way  to  the  tropic, 
or  to  the  90th  degree  of  longitude,  where  the  sun  and.  clock 
will  agree.  In  the  second  quadrant,  the  sun  will  come  to  the 
meridian  after  the  clock  has  marked  noon.  In  the  third  qua- 
drant the  sun  will  again  be  fast;  and,  in  the  fourth  quadrant, 
again  slow  of  clock. 

It  will  be  observed,  by  inspecting  the  figure,  that  what  the 
sun  loses  in  eastward  motion,  by  oblique  direction  near  the 
equator,  is  made  up,  when  near  the  tropics,  by  the  diminished 
distances  between  the  meridians. 

For  a  more  definite  understanding  of  this  matter,  we  givt 
the  following  table. 


EQUATION    OF    TIME. 


105 


Table  showing  the  separate  results  of  the  two  causes  for  the  equa-    CHAP.  v. 
(ion  of  time,  corresponding  to  every  f.fth  day  of  the  second 
years  after  leap  year  ;  bat  is  nearly  correct  for  any  year. 


1st  cause. 
Sun  slow 
of  Clock. 

2tl  cause. 
Sun  slow 
ofClock. 

1st  cause. 
Sun  fast. 

2d  cause. 
Sun  slow. 

m.  s. 

m.  s. 

m.  8. 

m.  s. 

January  5 
10 

0  41 
1  22 

5    8 
6  35 

July       1 

0    0 
0  40 

3  32 

5    8 

15 

2    2 

7  48 

12 

1  19 

6  35 

20 

2  41 

8  45 

17 

1  57 

7  48 

25 

3  19 

9  26 

22 

2  35 

8  45 

29 

3  56 

9  49 

28 

3  12 

9  26 

Feb.        3 

4  30 

9  53 

Aug.     2 

3  47 

9  49 

8 

5    2 

9  40 

7 

4  21 

9  53 

13 

5  32 

9    9 

12 

4  52 

9  40 

18 

5  39 

8  23 

17 

5  22 

9    9 

23 

6  24 

7  22 

22 

5  50 

8  23 

28 

6  45 

6    9 

.'       28 

6  14 

7  22 

March     5 

7    3 

4  46 

Sept.     2 

6  36 

6    9 

10 

7  18 

3  15 

7 

6  56 

4  46 

15 

7  29 

1  39 

12 

7  12 

3  15 

20 

7  37 

sun  fast 

17 

7  24 

1  39 

25 

7  42 

1  39 

23 

7  34 

sun  fast 

30 

7  42 

3  15 

28 

7  40 

1  39 

April       4 

7  40 

4  46 

Oct.       3 

7  42 

3  15 

9 

7  34 

6    9 

8 

7  40 

4  46 

14 

7  24 

7  22 

13 

7  34 

6    9 

19 

7  12 

8  23 

18 

7  24 

7  22 

24 

6  56 

9    9 

23 

7  12 

8  23 

30 

6  36 

9  40 

28 

6  56 

9    9 

May        5 

6  14 

9  53 

Nov.      2 

6  36 

9  40 

10 

5  50 

9  49 

7 

6  14 

9  53 

15 

5  22 

9  26 

12 

5  50 

9  49 

20 

4  52 

8  45 

17 

5  22 

9  26 

26 

4  21 

7  48 

'       22 

4  52 

8  45 

31 

3  47 

6  35 

27 

4  22 

7  48 

Ji'Me       5 

3  12 

5    8 

Dec.      2 

3  47 

6  35 

10 

2  35 

3  32 

7 

3  12 

5    8 

16 

1  57 

1  48 

12 

2  35 

3  32 

21 

1  19 

sun  slow 

17 

1  57 

1  48 

26 

0  40 

1  48 

21 

1  19 

sun  slow. 

26 

0  40 

1  48 

By  this  table,  the  regular  and  symmetrical  result  of  each  , 

cause  is  visible  to  the  eye ;  but  the  actual  value  of  the  equa-  preceding 
tion  of  time,  for  any  particular  day,  is  the  combined  results  tabl*« 
of  these  two  causes.     Thus,  to  find  the  equation  of  time  for 
the  5th  day  of  March,  we  look  at  the  table  and  find  that 


106  ASTRONOMY. 

CHAP,  v         The  first  cause  gives  sun  slow,  -     -     -       7m.  3  a. 
The  second,         "      sun  slow,  -     -     -       4   46 
Their  combined  result  (or  algebraic  sum)  is  11    49  slow. 

• 

That  is ,  the  sun  being  slow,  it  does  not  come  to  the  meridian 
until  llm.  49  s.  after  the  noon  shown  by  a  perfect  clock ;  but 
whenever  the  sun  is  on  the  meridian,  it  is  then  noon,  apparent 
time ;  and,  to  convert  this  into  mean  time,  or  to  set  the  clock, 
we  must  add  11  m.  49s. 

Use  of  the  By  inspecting  the  table,  we  perceive,  that  on  the  14th  of 
J^* '°  9  April  the  two  results  nearly  counteract  each  other ;  and  con- 
sequently the  sun  and  clock  nearly  agree,  and  indicate  noon 
at  the  same  instant.  On  the  2d  of  November  the  two  results 
unite  in  making  the  sun  fast ;  and  the  equation  of  time  is 
then  the  sum  of  6  36  and  9  40,  or  16  m.  16  s. ;  the  maximum 
result. 

The  sun  at  this  time  being  fast,  shows  that  it  comes  to  the 
meridian  16  m.  16  s.  before  twelve  o'clock,  true  mean  time ; 
or,  when  the  sun  is  on  the  meridian,  the  clock  ought  to  show 
11  h.  43  m.  44  s. ;  and  thus,  generally,  when  the  sun  is  fast,  we 
must  subtract  the  equation  of  time  from  apparent  time,  to  obtain 
mean  time  ;  and  conversely,  when  the  sun  is  slow. 

As  no  clock  can  be  relied  upon,  to  run  to  true  mean  time, 
or  to  any  exact  definite  rate,  therefore  clocks  must  be  fre- 
quently rectified  by  the  sun.  We  can  observe  the  apparent 
time,  and  then,  by  the  application  of  the  equation  of  time,  we 
determine  the  true  mean  time. 

A  table  for      (85.)  As  the  sun  has  a  particular  motion,  corresponding 

equation  of_t0  every  particular  point  on  the  ecliptic,  and,  at  the  same 

ing*  on^the  tmie»  *ne  particular  point  on  the  ecliptic  has  a  definite  rela- 

«m's    long!  tion  to  the  equator,  therefore  any  point,  as  S  (Fig.  22),  on 

be  the  ecliptic,  has  the  two  corrections  for  the  equation  of  time ; 

consequently  a  table  can  be  formed  for  the  equation  of  time, 

depending  on  the  longitude  of  the  sun;  and  such  a  table 

would  be  perpetual,  if  the  longer  axis  of  the  solar  orbit  did 

not  change  its  position  in  relation  to  the  equinoxes.     But  aa 

that  change  is  very  slow,  a  table  of  that  kind  will  serve  for 


PLANETARY    MOTIONS.  107 

n-any  years,  with  a  very  trifling  correction,  and  such  a  table    CHAP,  v 
is  to  be  found  in  many  astronomical  works. 

It  is  very  important  that  the  navigator,  astronomer,  and    Utility  of 
dock  regulator,  should  thoroughly  understand  the  equation  of 
time ;  and  persons  thus  occupied  pay  great  attention  to  it ; 
but  most  people  in  common  life  are  hardly  aware  of  its  ex- 
istence. 


CHAPTER    VI. 

THE   APPARENT   MOTIONS    OF   THE   PLANETS. 

(86.)  WE  have  often  reminded  the  reader  of  the  great  CHAP.  VL 
regularity  of  the  fixed  stars,  and  of  their  uniform  positions  in 
relation  to  each  other ;  and  by  this  very  regularity  and  con- 
stancy of  relative  positions,  we  denominate  them  fixed;  but 
there  are  certain  other  celestial  bodies,  that  manifestly  change 
their  positions  in  space,  and,  among  them,  the  sun  and  moon 
are  most  prominent. 

In  previous  chapters,  we  have  examined  some  facts  con-      Recapit* 
cerning  the  sun  and  moon,  which  we  briefly  recapitulate,  as  at!0n* 
follows : 

1.  That  the  sun's  distance  from  the  earth  is  very  great; 
but  at  present  we  cannot  determine  how  great,  for  the  want 
of  one  element  —  its  horizontal  parallax. 

2.  Its  magnitude  is  much  greater  than  that  of  the  earth. 

3.  The  distance  between  the  sun  and  earth  is  slightly  va- 
riable ;  but  it  is  regular  in  its  variations,  both  in  distance  and 
in  apparent  angular  motion. 

4.  The   moon  is  comparatively  very  near  the  earth;  its 
distance  is  variable,  and  its  mean  distance  and  amount  of 
variations  are  known.     It  is  smaller  than  the  earth,  although, 
to  the  mere  vision,  it  appears  as  large  as  the  sun. 

The  apparent  motions  of  both  sun  and  moon  are  always  in 
one  direction;  and  the  variations  of  their  motions  are  never 
far  above  or  below  the  mean.  other cefc* 

But  there  are  several  other  bodies  that  are  not  fixed  stars ;  tw  bodies. 


108  ASTRONOMY. 

C^__Z!'   an<^  altnough  no*  as  conspicuous  as  the  sun  and  moon,  hava 
been  known  from  time  immemorial. 

They  appear  to  belong  to  one  family;  but,  before  the  true 
system  of  the  world  was  discovei  ed,  it  was  impossible  to  give 
any  rational  theory  concerning  their  motions,  so  irregular 
and  erratic  did  they  appear;  and  this  very  irregularity  of 
their  apparent  motions  induced  us  to  delay  our  investigations 
concerning  them  to  the  present  chapter. 

Th«  plan.      jn  general  terms,  these  bodies  are  called  planets  —  and 

*"*"  there  are  several  of  recent  discovery  —  and  some  of  very 

recent  discovery;  but  as  these  are  not  conspicuous,  nor  well 

known,  all  our  investigations  of  principles  will  refer  to  the 

larger  planets,  Venus,  Mars,  Jupiter,  and  Saturn.     We  now 

commence  giving  some  observed  fads,  as  extracted  from  the 

Cambridge  astronomy 

The  morn.      (  87.)  "  There  are  few  who  have  not  observed  a  beautiful 


mg  and  even-  g£ar  jn  ^  wes^  a  little  after  sunset,  and  called,  for  this  rea- 
son, the  evening  star.  This  star  is  Venus.  If  we  observe  it 
for  several  days,  we  find  that  it  does  not  remain  constantly 
at  the  same  distance  from  the  sun.  It  departs  to  a  certain 
distance,  which  is  about  45°,  or  {th  of  the  celestial  hemi- 
sphere, after  which  it  begins  to  return  ;  and  as  we  can  ordi- 
narily discern  it  with  the  naked  eye  only  when  the  sun  is 
below  the  horizon,  it  is  visible  only  for  a  certain  time  imme- 
diately after  sunset.  By  and  by  it  sets  with  the  sun,  and 
then  we  are  entirely  prevented  from  seeing  it  by  the  sun's 
light.  But  after  a  few  days,  we  perceive,  in  the  morning, 
near  the  eastern  horizon,  a  bright  star  which  was  not  visible 
before.  It  is  seen  at  first  only  a  few  minutes  before  sunrise, 
and  is  hence  called  the  morning  star.  It  departs  from  the 
sun  from  day  to  day,  and  precedes  its  rising  more  and  more  ; 
but  after  departing  to  about  45°,  it  begins  to  return,  and 
rises  later  each  day  ;  at  length  it  rises  with  the  sun,  and  we 
cease  to  distinguish  it.  In  a  few  days  the  evening  star  again 
appears  in  the  west,  very  near  the  sun  ;  from  which  it  departs 
in  the  same  manner  as  before  ;  again  returns  ;  disappears  for 
a  short  time  ;  and  then  the  morning  star  presents  itself. 
These  alternations,  observed  without  interruption  for  more 


PLANETARY    MOTION  109 

than  2000  years,   evidently  indicate  that  the  evening  and  CHAP  TL 
morning  star  are  one  and  the  same  body.     They  indicate,  also, 
that  this  star  has  a  proper  motion,  in  virtue  of  which  it  oscil- 
lates about  the  sun,  vsometimes  preceding  and  sometimes  fol- 
lowing it. 

These  are  the  phenomena  exhibited  to  the  naked  eye ;  but 
the  admirable  invention  of  the  telescope  enables  us  to  carry 
our  observations  much  farther." 

(  88. )  On  observing  Venus  with  a  telescope,  the  irradiation    Th«  pha*«» 
is,  in  a  great  measure,  taken  away,  and  we  perceive  that  it  ° 
has  phases,  like  the  moon.     At  evening,  when  approaching  the 
sun,  it  presents  a  luminous  crescent,  the  points  of  which  are 
from  the  sun.     The  crescent  diminishes  as  the  planet  draws 
nearer  the  sun ;  but  after  it  has  passed  the  sun,  and  appears 
on  the  other  side,  the  crescent  is  turned  in  the  other  direction ; 
the  enlightened  part  always  toward  the  sun,  showing  that  it 
receives  its  light  from  that  great  luminary.     The  crescent 
now  gradually  increases  to  a  semicircle,  and  finally  to  a  full  0fVeni»  and 
circle,  ag  the  planet  again  approaches  the  sun ;  but,  as  the  it8  »pp»»nt 
crescent  increases,  the  apparent  diameter  of  the  planet  diminishes  /  have    corrtc 
and  at  every  alternate  approach  of  the  planet  to  the  sun,  the  spending 
phase  of  the  planet  is  full,  and  the  apparent  diameter  small;  c  ange8" 
and  at  the  other  approaches  to  the  sun,  the  crescent  diminishes 
down  to  zero,  and  the  apparent  diameter  increases  to  its 
maximum.     When  very  near  the  sun,  however,  the  planet  is 
lost  in  the  sunlight ;  but  at  some  of  these  intervals,  between 
disappearing  in  the  evening,  and  reappearing  in  the  morning, 
it  appears  to  run  over  the  sun's  disc  as  a  round,  black  spot ; 
giving  a  fine  opportunity  to  measure  its  greatest  apparent 
diameter.*     When  Venus  appears  full,  its  apparent  diameter 
is  not  more  than  10",  and  when  a  black  spot  on  the  sun,  it 
is  59". 8,  or  very  nearly  V.     Hence  its  greatest  distance  must 
be,  to  its  least  distance,  as  59". 8  to  10,  or  nearly  as  6  to  1. 

*  Astronomers  do  not  measure  the  apparent  diameters  of 
the  planets  by  the  process  described  for  the  sun  and  moon, 
because  they  pass  the  meridian  too  quickly.  Most  of  them  will 
pass  the  meridian  in  a  small  fraction  of  a  second.  They  use 


110  ASTRONOMY. 


vi.        (  89.  )   When   we  come  to  form  a  theory  concerning  the 
real  motion  of  this  planet,  we  must  pay  particular  attention 
to  the  fact,  that  it  is  always  in  the  same  part  of  the  heavens 
Venus  ai-  as  the  sun  —  never  departing  more  than  47°  on  each  side  of 
ivnys     near  ft  —  called  its  greatest  elongation.     In  consequence  of  being 
always  in  the  neighborhood  of  the  sun,  it  can  never  come  to 
the  meridian  near  midnight.     Indeed,  it  always  comes  to  the 
Greatest  meri(jiail  within  three  hours  20  minutes  of  the  sun,  and,  of 

<Jongation. 

course,  in  daylight.  But  this  does  not  prevent  meridian  ob- 
servations being  taken  upon  it,  through  a  good  telescope  ;  * 

a  micrometer,  which  is  two  spider  lines,  always  parallel,  near 
the  focus  of  a  telescope,  and  so  attached,  by  the  mechanism  of 
screws,  as  to  open  and  close  at  pleasure. 

To  understand  its  grade  of  adjustment,  bring  the  two  lines 
together,  so  as  to  form  one  line.  Then  take  any  object, 
whose  angular  diameter  is  known  at  that  time,  as  the  diame- 
ter of  the  sun,  and  open  the  lines  so  as  just  to  take  in  its 
disc,  counting  the  turns,  and  parts  of  a  turn  givqji  to  the 
index  screw  to  open  to  this  object.  From  this  we  can  com- 
pute the  angle  corresponding  to  one  turn,  or  to  any  part  of  a 
turn,  of  the  index  screw. 

Now  if  we  wish  to  measure  the  apparent  diameter  of  any 
planet,  bring  the  lines  together,  and  then  open  them,  just  to 
inclose  the  planet  4  and  the  number  of  turns,  or  the  part  of  a 
turn,  given  to  the  screw,  will  determine  the  result. 

This  may  not  be  the  exact  mechanism  of  every  micrometer, 
but  this  is  the  principle  of  their  construction. 

»  Perhaps  we  ought  to  have  informed  the  reader  before,  "  that  the 
stars  continue  visible  through  telescopes,  during  the  day,  as  well  as  the 
night  ;  and  that,  in  proportion  to  the  power  of  the  instrument,  not  only 
the  largest  and  brightest  of  them,  but  even  those  of  inferior  luster,  such 
as  scarcely  strike  the  eye,  at  night,  as  at  all  conspicuous,  are  readily 
found  and  followed  even  at  noonday,  —  unless  in  that  part  of  the  sky 
which  is  very  near  the  sun,  —  by  those  who  possess  the  means  of  point- 
ing a  telescope  accurately  to  the  proper  places.  Indeed,  from  the  bot- 
toms of  deep  narrow  pits,  such  as  a  well,  or  the  shaft  of  a  mine,  such 
bright  stars  as  pass  the  zenith  may  even  be  discerned  by  the  naked  eye; 
and  we  have  ourselves  heard  it  stated  by  a  celebrated  optician,  that  the 


PLANETARY    MOTION.  Ill 

and,  as  to  this  particular  planet,  it  is  sometimes  so  bright  as   CHAP.  vi. 
to  be  seen  by  the  unassisted  eye  in  the  daytime. 

(  90.)  Even  without  instruments  r.nd  meridian  observations,     Motion  ot 
the  attentive  observer  can  determine  that  the  motion  of  Venus,  Venus  in  re> 

spect  *o  the 

m  relation  to  the  stars,  is  very  irregular  —  sometimes  its  ,tars. 
motion  is  rapid  —  sometimes  slow  —  sometimes  direct — some- 
times stationary,  and  sometimes  retrograde;*  but  the  direct 
motion  prevails,  and,  as  an  attendant  to  the  sun,  and  in  its 
own  irregular  manner,  as  just  described,  it  appears  to  tra- 
verse round  and  round  among  the  stars. 

(  91.  )  But  Venus  is  not  the  only  planet  that  exhibits  the        Mercwy 
appearances  we  have  just  described.     There  is  one  other,  and  8imilar  in  a11 

8i)pca.ra.nccs 

only  one  —  Mercury  ;  a  very  small  planet,  rarely  visible  to  the  to  venas. 
naked  eye,  and  not  known  to  the  very  ancient  astronomers. 
Whatever  description  we  have  given  of  Venus  applies  to  Mer- 
cury, except  in  degree.  Its  variations  of  apparent  diameter 
are  not  so  great,  and  it  never  departs  so  far  from  the  sun ; 
and  the  interval  of  time,  between  its  vibrations  from  one  side 
to  the  other  of  the  sun,  is  much  less  than  that  of  Venus. 

(92.)  These  appearances  clearly  indicate  that  the  sun  must  be    A  concim- 
the  center,  or  near  the  center,  of  these  motions,  and  not  the  earth  ;  ilon* 
and  that  Mercury  must  revolve  in  an  orbit  within  that  of  Venus. 

So  clear  and  so  unavoidable  were  these  inferences,  that  even 
the  ancients  (who  were  the  most  determined  advocates  for 
the  immobility  of  the  earth,  and  for  considering  it  as  the 
principal  object  in  creation  —  the  center  of  all  motion,  etc.) 
were  compelled  to  admit  them ;  but  with  this  admission,  they 
contended,  that  the  sun  moved  round  the  earth,  carrying 
these  planets  as  attendants. 

(93.)  By  taking  observations  on  the  other  planets,  the  an-     The  appa 
cient  astronomers  found  them  variable  in  their  apparent  diam-  rent  diam* 

earliest  circumstance  which  drew  his  attention  to  astronomy,  was  the 
regular  appearance,  at  a  certain  hour,  for  several  successive  days,  of  a 
considerable  star,  through  the  shaft  of  a  chimney."— Herschd'a  Astro- 
nomy. 

*  In  astronomy,  direct  motion  is  eastward  among  the  stars  ;  station" 
ary  is  no  apparent  motion,  in  respect  to  the  stars  ;  and  retrograde  is  a 
westward  motion. 


112  ASTRONOMY. 

CHAP.  vi.    eters,  and  angular  motions;  so  much  so,  that  it  was  impossible 

ter»    of  the  to  reconcile  appearances  with  the  idea  of  a  stationary  point  of 

planets    are  observation ;  unless  the  appearances  were  taken  for  realities, 

and  that  was  against  all  true  notions  of  philosophy. 

The  planet  Mars  is  most  remarkable  for  its  variations ;  and 
the  great  distinction  between  this  planet  and  Venus,  is,  that 
it  does  not  always  accompany  the  sun;  but  it  sometimes,  yea, 
at  regular  periods,  is  in  the  opposite  part  of  the  heavens  from 
the  sun  —  called  Opposition  —  at  which  time  it  rises  about 
sunset,  and  comes  to  the  meridian  about  midnight. 
Th«  earth      The  greatest  apparent  diameter  of  Mars  takes  place  when 
uTr'ofiu  mo'  *ke  planet  *8  i°  opposition  to  the  sun,  and  it  is  then  17".l;  and 
tion.  its  least  apparent  diameter  takes  place  when  in  the  neighbor- 

hood of  the  sun,  and  it  is  then  but  about  4";  showing  that  the 
sun,  and  not  the  earth,  is  the  center  of  its  motion. 

Systematic  The  general  motion  of  all  the  planets,  in  respect  to  the 
ineguianties  g|-arg  js  direct ;  that  is,  eastward;  but  all  the  planets  that 
attain  opposition  to  the  sun,  while  in  opposition,  and  for  some 
time  before  and  after  opposition,  have  a  retrograde  motion  — 
and  those  planets  which  show  the  greatest  change  in  appa- 
rent diameter,  show  also  the  greatest  amount  of  retrograde 
motion  —  and  all  the  observed  irregularities  are  systematic  in 
their  irregularities,  showing  that  they  are  governed,  at  least, 
by  constant  and  invariable  laws.  If  the  earth  is  really  sta- 
tionary, we  cannot  account  for  this  retrograde  motion  of  tho 
planets,  unless  that  motion  is  real;  and  if  real,  why,  and 
how  can  it  change  from  direct  to  stationary,  and  from  station- 
ary to  retrograde,  and  the  reverse? 

Retrograde  But  if  we  conceive  the  earth  in  motion,  and  going  the  same 
motion  of  the  wgy  y^fa  the  planet,  and  moving  more  rapidly  than  the  planet, 
counted  for.  then  the  planet  will  appear  to  run  back  ;  thai  is,  retrograde. 

And  as  this  retrogradation  takes  place  with  every  planet, 
when  the  earth  and  planet  are  both  on  the  same  side  of  the 
sun,  and  the  planet  in  opposition  to  the  sun ;  and  as  these  cir- 
cumstances take  place  in  all  positions  from  the  sun.  it  is  a  suf- 
ficient explanation  of  these  appearances ;  and  conversely,  then, 
these  appearances  show  the  motion  of  the  earth. 

(94.)  When  a  planet  appears  stationary,  it  must  be  really 


PLANETARY   MOTION.  113 

so,  or  be  moving  directly  to  or  from  the  observer.     And  if  it   CHAP  vi. 
be  moving  to  or  from  the  observer,  that  circumstance  will  be  rianeu  ner- 
indicated  by  the  change  in  apparent  diameter;  and  observa-  •'•tationajy» 
lions  confirm  this,  and  show  that  no  planet  is  really  station- 
ary, although  it  may  appear  to  be  so. 

(95.)  If  we  suppose  the  eartb.  to  be  but  one  of  a  family  of  The  earth  « 
bodies,  called  planets  —  all  circulating  round  the  sun  at  dif-  P1&Mt' 
ferent  times  —  in  the  order  of  Mercury,  Venus,  Earth,  Mars 
(omitting  the  small  telescopic  planets),  Jupiter,  Saturn,  Her- 
schel,  or  Uranus,  «?e  can  then  give  a  rational  and  simple  ac- 
count for  every  appearance  observed,  and  without  discussing 
the  ancient  objections  to  the  true  theory  of  the  solar  system, 
we  shall  adopt  it  at  once,  and   thereby  save  time  and  labor, 
and  introduce  the  reader  into  simplicity  and  truth. 

(96.)  The  true  solar  system,  as  now  known  and  acknow- 
ledged,  is  called  the  Copernican  system,  from  its  discoverer, 
Copernicus,  a  native  of  Prussia,  who  lived  some  time  in  the  tem. 
fifteenth  century. 

But  this  theory,  simple  and  rational  as  it  now  appears,  and  Lost  and  re. 
capable  of  solving  every  difficulty,  was  not  immediately  adop-  ¥lv*d* 
ted ;  for  men  had  always  regarded  the  earth   as  the   chief 
object  in  God's  creation ;  and  consequently  man,  the  lord  of  crea 
tion,  a  most  important  being.    But  when  the  earth  was  hurled 
from  its  imaginary,   dignified  position,   to   a   more   humble 
place,  it  was  feared  that  the  dignity  and   vain  pride  of  man 
roust  tall  with  it ;  and  it  is  probable  that  this  was  the  root 
of  the  opposition  to  the  theory. 

So  violent  was  the  opposition  to  this  theory,  and  so  odious  Galileo  and 
would  any  one  have  been  who  had  dared  to  adopt  it,  that  it 
appears  to  have  been  abandoned  for  more  than  one  hundred 
years,  and  was  revived  by  Galileo  about  the  year  1620,  who, 
to  avoid  persecution,  presented  his  views  under  the  garb  of  a 
dialogue  between  throe  fictitious  persons,  and  the  points  left 
undecided. 

But  the  caution  of  Galileo  was  not  sufficient,  or  his  dia- 
logue was  too  convincing,  for  it  woke  up  the  sacred  guardians 
of  truth,  and  he  was  forced  to  sign  a  paper  denouncing  the 
theory  as  heresy,  on  the  pain  of  perpetual  imprisonment. 
8 


114 


ASTRONOMY, 


CHAP.  VL  But  this  is  a  digression.  With  the  history  of  astronomy,  an 
interesting  as  it  may  be,  we  design  to  have  little  to  do,  and 
to  proceed  only  with  the  science  itself. 


Distinction 
between  in- 
terior and  su- 
perior plan- 
ets. 


CHAPTER    VII. 

FIRST   APPROXIMATIONS     TO    THE    RELATIVE     DISTANCES   OF   THB 
PLANETS  FROM  THE  SUN.       HOW  THE  RESULTS  ARE  OBTAINED. 

(97.)  BEING  convinced  of  the  truth  of  the  Copernican 
system,  the  next  step  seems  to  be,  to  find  the  periodical  times 
of  the  revolutions  of  the  planets,  and  at  least  their  relative 
distances  from  the  sun. 

Mercury  and  Venus,  never  coming  in  opposition  to  the  sun, 
but  revolving  around  that  body  in  orbits  that  are  within  that 
°f  tne  earfcn'  are  called  inferior  planets. 

Those  that  come  in  opposition,  and  thereby  show  that 
their  orbits  are  outside  of  the  earth,  are  called  superior 
planets. 

We  shall  show  how  to  investigate  and  determine  the  posi- 
tion of  one  inferior  planet ;  and  the  same  principles  will  be 
sufficient  to  determine  the  position  of  any  inferior  planet. 

It  will  be  sufficient,  also,  to  investigate  and  determine  the 
orbit  of  one  superior  planet ;  and  if  that  is  understood,  it  may 
be  considered  as  substantially  determining  the  orbits  of  all 
the  superior  planets ;  and  after  that,  it  will  be  sufficient  tv 
state  results. 

For  materials  to  operate  with,  we  give  the  following  table 
of  the  planetary  irregularities  ( so  called  )  drawn  from  obser- 
vation : 


Planet*. 

Greatest 
Apparent 
Diameters. 

Least 

Apparent 
Diameters. 

Angular     Dist. 
from  Sun  at  the 
instant  of  being 
stationary. 

Mean  arc  of 
Retrogradation. 

Mercury. 
Venus. 
Earth. 
Mars. 
Jupiter. 
Saturn. 
Uranus. 

11.3 
59.6 

17*1 

44.5 
20.1 
4.1 

5.0 
9.6 

3.6 
30.1 
16.3 
3.7 

18  00 
28  48 

136  48 
115  12 
108  54 
103  30 

u       f 

13  30 
16  12 

16  12 
9  54 
6  18 
3  36 

PLANETARY    MOTION. 


115 


Planets. 

Mean  Duration  of  the  Retro- 
grade Motion. 

Mean  Duration  of  the  Synodic 
Revolution,  or  interval  between 
two  successive  oppositions. 

Mercury. 

23  days. 

118   days. 

Venus. 

42      " 

584      " 

Earth. 

Mars. 

73      " 

780     " 

Jupiter. 

121      " 

399      « 

Saturn. 

139      " 

378     « 

Uranus. 

151      " 

370     « 

CHAP.  VIL 


In  the  preceding  table,  the  word  mean  is  used  at  the  head     why  the 
of  several  columns,  because  these  elements  are  variable  —  word    at:AK 

,  .  .  should  be 

sometimes  more  and  sometimes  less,  than  the  numbers  here  used. 
given  —  which  indicates  that  the  planets  do  not  revolve  in  cir- 
cles round  the  sun,  but  most  probably  in  ellipses,  like  the  orbit 
of  the  earth. 

On  the  supposition,  however/  that  the  planets  revolve  in 
circles  (  which  is  not  far  from  the  truth  ),  the  greatest  and 
least  apparent  diameters  furnish  us  with  sufficient  data  to 
compute  the  distances  of  the  planets  from  the  sun  in  relation 
to  the  distance  of  the  earth,  taken  as  unity* 

(98.)  In  addition  to  the  facts  presented  in  the  preceding  The  eionga. 
table,  we  must  not  fail  to  note  the  important  element  of  the  tion*of  Mer- 
dongaiions  of  Mercury  and  Venus.     This  term  can  be  applied  nus> 
to  no  other  planets. 

It  is  very  variable  in  regard  to  Mercury  —  showing  that  This  element 
the  orbit  of  that  planet  is  quite  elliptical.      The  variation  is 
much  less  in  regard  to  Venus,  showing  that  Venus  moves  shows. 
round  the  sun  more  nearly  in  a  circle. 

The  least  extreme  elongation  of  Mercury  is    -     17°  37'. 

The  greatest  "  "  "          is    -     28°     4'. 

The  mean  (or  the  greatest  elongation  when 
both  the  earth  and  planet  are  at  their 
mean  distances  from  the  sun  )  is  -  -  -  22°  46'. 

The  least  extreme  elongation  of  Venus  is     -     44°  58'. 

The  greatest  "  "  "         is     -     47°  30'. 

The  mean   (or  at  mean  distances),          is     -     46°  30'. 

The  least  extremes  must  happen  when  the  planet  is  in  ita 
perigee  and  the  earth  in  its  apogee,  and  the  greatest  when 
the  earth  is  in  perigee  and  the  planet  in  apogee;  but  it  is 


and 


116 


A3TRONOMY. 


CHAH.  vii.  very  seldom  that  these  two  circumstances  take  place  at  the 

same  time. 

HOW   to      Relying  on  these  facts  as  established  by  observations,  wo 
can  easily  deduce  the  relative  orbits  of  Mercury  and  Venus. 

Let  .V  (Fig.  23)  re- 
present the  sun,  E  the 
earth,  V  Venus. 

Conceive  the  planet 
to  pass  round  the  sun 
in  the  direction  of  A 

^7  B. 

The  earth  moves  also 
in  the  same  direction, 
but  not  so  rapidly  as 
Venus. 

Now  it  is  clearly  evi- 
dent, from  inspection, 
that  when  the  planet  is 
passing  by  the  earth,  as 
at  B,  it  will  appear  to 
pass  along  in  the  hea- 
vens in  the  direction  of 
m  to  n.  But  when  the  planet  is  passing  along  in  its  orbit,  at 
A,  and  the  earth  about  the  position  of  E,  the  planet  will 
appear  to  pass  in  the  direction  of  n  to  w.  When  the  planet 
is  at  V,  as  represented  in  the  figure,  its  absolute  motion  is 
nearly  toward  the  earth,  and,  of  course,  its  appearance  is 
nearly  stationary. 
What  to  jj.  'ls  l(j)SOluieiy  stationan/  only  at  one  point,  and  even  then 

understand  .          .         , 

by    station-  but  for  a  moment ;  and  that  point  is  where  its  apparent  nio- 
»ry.  tion  changes  from  direct  to  retrograde,  and  from  retrograde 

to  direct;  which  takes  place  when  the  angle  SE  V  is  about 
29  degrees  on  each  side  o*'  the  line     /'. 

When  the  line  E  V  touches  the  circumference  A  VB,  the 
angle  £  E  \\  or  angle  of  elongation,  is  then  greatest ;  and  the 
triangle  SE  Vis  right  angled  at  V;  and  if  SE  is  made  ra- 
dius, S  V  will  be  the  sine  of  the  angle  SE  V. 

Bu*«  the  line  S  E  is  assumed  equal  to  unity,  and  then  S  V 


PLANETARY    MOTION.  jj7 

will  be  thfs  natural  sine  of  46°  20',  and  can  be  taken  out  of   CHAP  vu 
any  table  of  natural  sines ;  or  it  can  be  computed  by  loga- 
rithms,  and  the  result  is  .72336. 

For  the  planet  Mercury,  the  mean  of  the  same  angle  is 
*2"2°  46';  and  the  natural  sine  of  that  angle,  or  the  mean  radius 
of  the  planet's  orbit,  is  .38698. 

Thus  we  have  found  the  relative  mean  distances  of  three 
planets  from  the  sun,  to  stand  as  follows: 

Mercury,      -  •  0.38698 

Venus,     -         -  -    0.72336 

Earth,  -         -  1.00000 

(  99. )  If  the  orbits  were  perfect  circles,  then  the  angle    Thfl  °'bit§ 
SE  V,  of  greatest  elongation,  would  always  be  the  same;  °nd    Venn* 
but  it  is  an  observed  fact  that  it  is  not  always  the  same ;  not  circles, 
therefore  the  orbits  are  not  circles ;  and  when  S  V  is  least, 
and  S  E  greatest,  then  the  angle  of  elongation  is  least ;  and 
conversely,  when   S  V  is  greatest  and  S  E  least,  then  the 
angle  of  elongation  is  the  greatest  possible ;  and  by  observing 
in  what  parts  of  the  heavens  the  greatest  and  least  elongations 
take  place,  we  can  approximate  to  the  positions  of  the  longer 
axis  of  the  orbits. 

(  100. )  By  means  of  the  apparent  diameters,  we  can  also      Comimta- 
find  the  approximate  relations  of  their  orbits.     For  instance,  J^0  °r '" 
when  the  planet  Venus  is  at  B,  and  appears  on  the  sun's  rent  diamo- 
disc,  its  apparent  diameter  is  59".6 ;  and  when  it  is  at  A,  or  ter$ 
as  near  A  as  can  be  seen  by  a  telescope,  its  apparent  diame- 
ter is  9".6.     Now  put 

SB=xj     then     EB=l—x,     and     AE=\-\-x. 

By  Art.  66,  l—x     :     1+x    :  :     96     :     696; 

Hence,     -     -     -     -     *=0.72254. 

By  a  like  computation,  the  mean  distance  of  Mercury  from 
tne  sun  is  0.3864. 

(101.)  To  determine  the  mean  relative  distances  of  the 
superior  planets  from  the  sun,  we  proceed  as  follows : 

Let  S  (Fig.  24)  represent  the  sun,  E  the  earth,  and  J/"one 
of  the  superior  planets,  say  Mars.  It  is  easy  to  decide,  from 
observation,  when  the  planet  is  in  opposition  to  the  sun. 


118 


ASTRONOMY. 


CHAP.  V!i.  Fig.  24  This  gives  the  position 

of  S,  E,  and  M,  in  one 
right  line,  in  respect 
to  longitude.  Now  by 
knowing  the  true  angu- 
lar motion  of  the  earth 
about  the  sun  (73),  and 
the  mean  angular  mo- 
tion of  the  planet,  *  v:o 
can  determine  the  anglt1 
mSe,  corresponding  to 
any  definite  future  time  ; 
for,  by  the  motion  of  the 
earth  round  the  sun,  we 
can  determine  the  angle 
E Se;  and  by  the  mo- 
tion of  the  planet  in  the 
same  time,  we  can  determine  the  angle  M S  m  ;  and  the  dif- 


Jjy  means  of  apparent   diameters,  we   can  determine  the 
values  of  the  orbit.     When  the  planet  is  in  opposition  to  the 
the  sun  de- sun,  at   E( Fig.  24),   measure  its  apparent   diameter;  and, 
tera:ned  by  after  a  defimte  time,  when  the  earth  is  at  e,  measure  the  ap- 
tion   limits  parent  diameter  again,  and  observe  the  angle  S  cm.     Pro- 
apparent  <iia- duce  Se  to  n.     Then,  by  the  apparent   diameters,  we  have 
the  proportion  of  e  m  and  e  n  (e  n  is  the  same  as  E  M,  brought 
to  this  position);  and  in  the  triangle  e  mn  we  have  the  pro- 
portion between  the  two  sides  and  the  included  angle  m  e  n. 
These   are  sufficient  data  to  determine  the  angles  enm  and 
emn;  and  their  difference  is  the  angle  Sme.     Now  we  can 
determine  the  side  S  m,  of  the  triangle  Sme,  and  the  triangle 
Sem  is  completely  known.     Subtract  the  angle  e  Sm  from 
the  whole  angle  e  S  M,  and  the  angle  M  Sm  is  left.     That 
is,  while  the  earth  is  describing  the  angle  E  Se,  the  planet 
describes  the  angle  MSm.     Put  P  for  the  periodical  revo- 

*  Here  we  anticipate  a  little  ;  for  we  have  not  shown  how  to  deter- 
mine the  periodical  time  of  revolution  from  observation :  but  this  i» 
shown  in  a  future  chapter,  and  in  the  above  text  note 


PLANETARY    MOTION.  119 

ference  of  these  two  angles  is  the  angle  m  S  e.     By  direct  CHAP,  vn 
observation   at  e,  we  determine  the  angle  Sem;  and   two 
angles,  and  the  side  S  e,  of  the  triangle  Sme,  are  sufficient  to 
determine  the  side   Sm,  the  value   sought.     The   triangle 
gives  the  following  proportion : 

.  sin.  Sem 

sm.  Sme    :     1     ::     sm.  Sem    :     Sm=- — ~— . 

sm.  Sme 

Thio  is  a  general  solution,  for  any  superior  planet ;  but  the     why  tho 
result  is  only  approximate ;  for,  until  we  know  the  eccentri-  resn!t  is  &P- 
city  of  the  orbit  in  question,  and  the  part  of  the  orbit  in  F< 
which  the  planet  then  is,  we  cannot  accurately  know  the 
angle  MSm. 

lution  of  the  planet;  then,  on  the  supposition  of  uniform 
motion,  we  have 

SLYG  MSm    :    &YG  E8e    ::     365i     :     P 

In  this  proportion  the  two  arcs  are  known,  and  from  thence 
P  becomes  known ;  and  thus,  we  perceive,  thai  by  the  variations 
of  the  apparent  diameter  of  a  planet,  we  can  determine  its  rela- 
tive distance  from  the  sun,  and  its  periodical  revolution. 

We  give  the  following  hypothetical  example,  for  the  pur- 
pose of  further  illustration. 

The  apparent  diameter  of  Mars,  when  in  opposition  to  the  sun,  A  prob.en 
was  observed  to  be  17". .1.  One  hundred  and  eleven  days  after- 
ward, when  the  earth  had  passed  over  110°  of  its  orbit,  the  appa- 
rent  diameter  of  Mars  was  again  observed,  and  found  to  be  1"  A, 
and  its  angular  position,  in  longitude,  was  87°  from  the  sun. 
From  these  data,  it  is  required  to  find  the  relative  approximate 
distance  of  the  planet  from  Die  sun,  and  the  approximate  time  of 
its  revolution  round  the  sun. 

From  these  data  we  have  the  angle  MSn=llO°,  Se  m=       its  soit 
«7° ;  therefore  n  e  m=93°.  *™  ~  Fig" 

By  the  observed  apparent  diameter,  we  have  EM  to  em  ~ 
as  7".4  to  17".l;  but  EM=en,  therefore 

en    :    em    ::     74     :     171. 

In  the  triangle  nem  we  can  take  en=74,  and  2?w=171, 
for  the  purpose  merely  of  finding  the  angles.  Then,  by  trigo- 
nometry, we  have 


120 


ASTRONOMY 


CH*P._VII.       ( 102.)  By  a  perusal  of  the  last  text  note,  it  will  be  seen, 

Rczniu  by  those  even  who  are  not  expert  mathematicians,  that  it  is 

from    varia-  not  djfficuifc  fa  decide  upon  the   relative  distances   of  the 

tjoaa    in  ap- 
parent   dia-  planets  from  the  sun,  by  observing  their  changes  in  apparent 

diameter,  as  seen  from  the  earth.  Such  observations  have 
been  often  made,  and  the  following  table  shows  the  results; 
which  are  compared  with  the  results  deduced  from  Kepler's 
Third  Law.* 


Planets. 

Deduced  from  appa- 
rent Diameters. 

From  Kepler's 
Law. 

Difference  or 
Error. 

Mercury 
Venus  . 
Earth  .  . 
Mart  .. 

0.3b6400 
0.722540 
1.000000 
1.533333 

0.387098 
0.723331 
1.000000 
1.523692 

—.000699 
—.000791 

4-.009641 

Jupiter. 
Saturn  . 
Uranus. 

5.180777 
9.579000 
19.500000 

5.202776 
9.53S786 
19.182390 

—  .021999 
4-.040214 
-f.317610 

Text  note 

continued. 


87° 

171+74  :  171—74  : :  tan.  -^~  :  tan.  1,  difference  be- 
tween the  angle  n  and  nme. 

That  is,  -   245     :    97     : :    tan.  43°  30'    :    tan  1  Smt. 
Whence,      Sme=£\?  11'.     Now  in  the  triangle  Sme, 
sin.  41°  11'    :     1    :  :    sin.  87°    :     £»i=1.517. 
Secondly,  as  the  angle  £m*=41°  11'  and  Sem  87C,  there- 
fore, -    -    7n&=51°  49',    and    MSm  58°  11'. 

But  the  times  of  revolution,  between  any  two  planets,  must 
be  inversely  as  the  angles  they  describe  in  the  same  time ; 
the  greater  the  angle,  the  shorter  the  periodic  time;  and 
therefore  if  we  put  P  to  represent  the  periodical  revolution 
of  Mars,  we  shall  have 

58T2T  :  110  : :  365i  :  P.     Hence  7>=690f  days. 

The  true  time  is  686.97964;  showing  an  error  of  a  little 
more  than  three  days ;  but  this  is  not  a  great  error,  consider- 
ing the  remoteness  of  the  data,  and  the  want  of  minuteness  .and 
unity  in  the  supposed  observations.  Our  object  is  only  to 
teach  principles ;  not,  as  yet,  to  establish  minute  results. 


»  A  principle  to  be  explained  in  Physical  Astronomy. 


PLANETARY    MOTION.  12' 

The  distances  drawn  from  Kepler's  law,  are  considered  CHAP.  vn. 
more  accurate  than  conclusions  drawn  from  most  other  con-  Why  tha 
siderations ;  and  it  is  rather  remarkable  that  these  deduc-  result*  from 

/»  ,1  T  11  Ai  ••       apparent    di- 

tions  iroin  the  apparent  diameters  agree  as  well  as  they  ao,  ameter,  can. 
owing  to  the  difficulty  of  settling  the  exact  apparent  diain-  not  be  relied 
eter,  by  observation.  Take  the  apparent  diameter  of  lira- 
nus,  for  example,  3".7  and  4".l,and  change  either  of  them 
y1^  of  a  second,  and  it  will  make  a  great  difference  in  the 
deduced  result. 


CHAPTER    VIII. 

H  ETHODS   OF   OBSERVING    THE   PERIODICAL   REVOLUTIONS   OF  THE 
PLANETS,    AND   THEIR   RELATIVE  '  DISTANCES    FROM   THE   SUN. 

(  103.)  THE  subject  of  this  chapter  will  be  to  explain  the  CHAP,  vm. 
principles  of  finding  the  periodical  revolutions  of  the  planets    why  direct 


around  the  sun.  If  observers  on  the  earth  were  at  the  re  nt  to  th« 
center  of  motion,  they  could  determine  the  times  of  revo-  point 
lution  by  simple  observation.  But  as  the  earth  is  one  of  the 
planets,  and  all  observers  on  its  surface  are  carried  with  it, 
the  observations  here  made  must  be  subjected  to  mathemati- 
cal corrections,  to  obtain  true  results  ;  and  this  was  an  impos- 
sible problem  to  the  ancients,  as  long  as  they  contended  for  a 
stationary  earth. 

If  the  observer  could  view  the  planets  from  the  center  of    TWO  i 
the  sun,  he  would  see  them  in  their  true  places  among  the  ^  P 
stars  —  and  there  are  only  two  positions  in  which  an  observer 
on  the  earth  will  see  a  planet  in  the  same  place  as  though  he 
viewed  it  from  the  center  of  the  sun,  and  these  positions  are 
conjunction  and  opposition. 

Thus,  in  Fig.  24,  when  the  earth  is  at  E,  and  a  planet  at 
M,  the  planet  is  in  opposition  to  the  sun  ;  and  it  is  seen  pro- 
jected among  the  stars  at  the  same  point,  whether  viewed 
from  S  or  from  E. 

In  Fig.  23,  if  the  planet  is  at  B,  or  A,  it  is  said  to  be  in 
conjunction  with  the  sun;  but  a  conjunction  cannot  be  ob-  »«rv«d. 


A&lRONOMY. 

viii.  served  OK  account  of  the  brilliancy  of  the  sun,  unless  it  be  the 
two  planets,  Mercury  and  Venus,  and  then  only  when  they 
pass  directly  before  the  face  of  the  sun,  and  are  projected  on 
its  surface  as  a  black  spot.    Such  conjunctions  are  culled  transits. 
( 104.)  All  the  planets  move  around  the  sun  in  the  same 
Revolution  direction,  and  not  far  from  the  yarne  plane,  and  the  rudest 

of     inferior      ^  mogt  carejess  observations  show  that  those  planets  near- 
planets  less, 

and  of  supe-  est  the  sun,  perform  their  revolutions  in  shorter  periods  than 
tint   planets  tnoge  more  remote.     From  this,  we  decide  at  once  that  the 

g renter    than 

a  year.  mean  angular  motion  of  all  the  superior  planets  is  less  than 
the  mean  angular  motion  of  the  earth  in  its  orbit;  and  the 
mean  angular  motion  of  the  inferior  planets,  as  seen  from 
the  sun,  is  greater  than  the  mean  motion  of  the  earth. 

( 105.)  The  time  that  any  planet  comes  in  opposition  to 
Times  of  the  sun,  can   be  very  distinctly  determined  by  observation, 
opposition     jtg  longitude  is  then  180  degrees  from  the  longitude  of  tho 
served          sun,  and  comes  to  the  meridian  nearly  or  exactly  at  midnight. 
If  it  is  a  little  short  of  opposition  at  the  time  of  one  obser- 
vation, and  a  little  past  at  another,  the  observer  can  propor- 
tion to  the  exact  time  of  opposition,  and  such  time  can  be 
definitely  recorded  —  and  by  such  observation,  we  have  the 
true  position  of  the  planet,  as  seen  from  the  sun.     Another 
pi     25  opposition  of  the  same  kind  and 

of  the  same  planet,  can  be  ob- 
served and  recorded. 

The  elapsed  time  between  two 

Synodic*]  ^^^^^^^HNHRBH^H|  such  oppositions  is  called  the  sy- 

nodical  revolution  of  the  planet. 
We  note   the  time   that   a 
„  lt .  , ........          .  ^-;;,     ..,,  .r.,._  planet  is  in  opposition  to  tho 

hr  motion  oi  K&O^S^-i  J¥.l  ^1 '.'.,  W'SJ   SUn'       ^n611  ^'  ^  an(*  -^"are  m 

planet?  ttS^^^^S  P^S^^^*|H  one  P^ane  as  represented  in  Fig. 

5«       If    the  Pla"Ct    M  Sh°u^ 

synodical      ^y?:^^^  J^  iWfcl^I^S '•  remain  at  rest  while  the  earth 
revolution..   |y ®|^ v ig|||^|  E   made  its  revolution,  then 

the  synodical  revolution  would 
be  the  same  as  the  length  of 
our  year.     But  all  the  planets  move  in  the  same  directior   vi 


PLANETARY    MOTION.  123 

the  earth;  and  therefore  the  earth,  after  making  a  revolu-  CHip.vni 
tion,  must  pass  onward  and  employ  additional  time  to  over- 
take the  planet ;  and  the  more  rapidly  the  planet  moves,  the 
longer  time  it  will  require.  Hence,  in  case  two  planets  have 
but  a  small  difference  in  angular  motion,  their  synodical  pe-  General  coo- 
riod  must  be  proportionately  long.  The  planet  Jupiter  ^derations 
moves  about  31°  in  its  orbit  in  a  year;  and  therefore,  after 
one  opposition,  the  earth  is  round  to  the  same  point  in  365} 
days,  and  to  gain  the  31°  requires  about  32  days  more ;  hence 
the  synodical  revolution  of  Jupiter  must  be  about  397  days, 
by  this  vei-y  rough  and  imperfect  computation.  By  inspect- 
ing the  table  on  page  105,  we  perceive  that  the  mean  synodi- 
cal revolution  of  Jupiter  is  399  days,  and  this  observed  fact 
shows  us  that  Jupiter  passes  over  about  31°  in  a  year,  and  of 
course  its  revolution  must  be  a  little  less  than  12  years;  and 
by  the  same  considerations,  we  can  form  a  rough  estimate  of 
the  periodical  revolutions  of  all  the  planets. 

(  106.)  The  general  principle  being  understood,  we  may 
now   be    more  scientific.       The  mean  motion    of  the    earth  Computatio» 
in  its  orbit  is  very  accurately  known.     Represent  its  daily  *?  determil)* 
motion  by  a.      The  angular  motion  of  the  planet  (  any  supe-  xniai  motioa 
rior  planet  that  maybe  under  consideration)  is  unknown ;' Ttbee*rth> 
therefore,  represent  its  daily  motion  by  x.     Let  the  angle  E 
SC  represent  a,  and  the  angle  MS  m  represent  x\  then  the 
angle  m  SCor  (a — x)  will  represent  the  daily  angular  advance 
of  the  earth  over  the  planet ;  and  as  many  times  as  the  an- 
gle m  SC  is  contained  in  360°,  will  be  the  number  of  days  in 

a.  synodical  revolution.      Therefore, =   the  observed 

a  —  x 

time  of  a  synodical  revolution ;   and  by  taking  the  times  from 
the  table  (page  105),  we  have  the  following  equations: 

Mart.  Jupiter.  Saturn.  Uranus. 

_.    i  __          360 

a  — x 


d 

*  These  equations  correspond  to  the  general  equation  f=ss_ — _    In 

ltobinson*s  Algebra,  page  105,  University  edition. 


124  ASTRONOMY 


.  vin.  The  value  of  a  is  59'  8",  and  then  a  solution  of  these  sev- 
era:  equations  gives  the  mean  angular  motion,  per  day,  of  the 
several  planets,  as  follows  : 

Mars.  Jupiter.  Saturn.  Uranus. 

31'  27"  4'59".4  r  59".5  45".3 

Times  of  Dividing  the  whole  circle  360°  by  the  mean  daily  motion 
dTrivecTfrom  °^  eac^  planet/>  w^  g^ve  their  respective  times  of  revolution, 
the  angular  and  the  following  are  the  results  : 

Mars,  Jupiter.  Saturn.  lL»r>us. 

687  days.       4331  days.        10840  days.       28610  days. 
(  106.)  For  the  inferior  planets,  Mercury  and  Venus,  we 
have  the  same  principle,  only  making  x  greater  than  a,  and 

For  Mercury.  For  Venus. 

8«0_ 


*=4°2'  11";  s=l°36'7". 

Mean  an-      These  diurnal  angular  motions  correspond  to  89  days  for 
guiar  motion  ^    revolution  of  Mercury,  and  224.8  days  for  the  revolution 

ofthe  inferior  *  '  J 

planets,  and  of  Venus.     All  these  results  are,  of  course,  understood  as 

their  revoiu-  grgt  approximations,  and   accuracy  here  is  not  attempted. 

«e  «un.        We  are  only  showing  principles  ;  and  it  will  be  noticed,  that 

the  times  here  taken  in  these  considerations,  are  only  to  the 

nearest  days  ,  and  not  fractions  of  a  day,  as  would  be  necessary 

for  accurate  results.     By  this  method  accuracy  is  never  at- 

tempted, on  account  of  the  eccentricity  of  the  orbits.     No 

two   synodical  revolutions  are  exactly  alike  ;  and  therefore 

it  is  very  difficult  to  decide  what  the  real  mean  values  are. 

(107.)  To  obtain  accuracy,  in  astronomy,  observations 
must  be  carried  through  a  long  series  of  years.  The  follow- 
ing is  an  example  ;  and  it  will  explain  how  accuracy  can  be 
attained  in  relation  to  any  other  planet. 

On  the  7th  of  November,  1631,  M.  Cassini  observed  Mer- 
cury passing  over  the  sun  ;  and  from  his  observations  then 
taken,  deduced  the  time  of  conjunction  to  be  at  7  h.  50  m.,  mean 
time,  at  Paris,  and  the  true  longitude  of  Mercury  44°  41'  35". 
Comparing  this  occultation  with  that  which  took  place  in 

1723>  the  true  time  of  conJunction  was  November  9th,  at  5  h. 
29m.,  P.  M.,  and  Mercury's  longitude  was  46°  47   20". 


PLANETARY    MOTION.  125 

The  elapsed  time  was  92  years,  2  days,  9  h.  39  m.  Twenty-  CHAP.  VHI. 
two  of  these  years  were  bissextile  ;  therefore  the  elapsed  time  Of  years,  to 
was  (92  X  365)  days,  plus  24  d.  9  h.  39  ra.  «ecw« 

In  this  interval,  Mercury  made  382  revolutions,  and  2°  5'  n 
45"  over.     That  is,  in  33604.402  days,  Mercury  described 
137522.095826  degrees;  and  therefore,  Toy  division,  we  find 
that  in  one  day  it  would  describe  4°.0923,  at  a  mean  rate. 

Thus,  knowing  the  mean  daily  rate  to  great  accuracy,  the 
mean  revolution,  in  time,  must  be  expressed  by  the  fraction 

or,  87.9701  days,  or  87  days  23  h.  15m.  57  s. 


(  108.  )  The  following  is  another  method  of  observing  the        Another 
periodical  times  of  the  planets,  to  which  we  call  the  student's  ™beser°in  th, 

Special  attention.  periodical  re- 

The  orbits  of  all  the  planets  are  a  little  inclined  to  the  volutions  of 

1          ,  the  planets. 

plane  of  the  ecliptic. 

The  planes  of  all  the  planetary  orbits  pass  through  the 
center  of  the  sun  ;  the  plane  of  the  ecliptic  is  one  of  them, 
and  therefore  the  plane  of  the  ecliptic  and  the  plane  of  any 
other  planet  must  intersect  each  other  by  some  line  passing 
through  the  center  of  the  sun.  The  intersection  of  two  planes 
is  always  a  straigJit  line.  (See  Geometry.) 

The  reader  must  also  recognize  and  acknowledge  the  fol- 
lowing principle  : 

That  a  body  cannot  appear  to  be  in  the  plane  of  an  observer, 
unless  it  really  is  in  that  plane. 

For  example  :  an  observer  is  always  in  the  plane  of  his 
meridian,  and  no  body  can  appear  to  be  in  that  plane  unless 
it  really  is  in  that  plane  ;  it  cannot  be  projected  into  or  out  of 
that  plane,  by  parallax  or  refraction. 

Hence,  when  any  one  of  the  planets  appears  to  be  in  the 
plane  of  the  ecliptic,  it  actually  is  in  that  plane  ;  and  let  the 
time  be  recorded  when  such  a  thing  takes  place. 

The  planet  will  immediately  pass  out  of  the  plane,  because      what   » 
the  two  planes  do   not  coincide.     Passing  the  plane  of  the  ln('ant      *" 
ecliptic  is  called  passing  the  node.     Keep  track  of  the  planet 
until  it  comes  into  the  same  plane  ;  that  is,  crosses  the  other 
node  :  in  this  interval  of  time  the  planet  has  described  just 


126  ASTRONOMY. 

CHAP.  vm.  180°,  as  seen  front,  the  sun  (unless  the  nodes  themselves  are 
Two  nod.,  in  motion,  which  in  fact  they  are ;  but  such  motion  is  not 
180   degree*  sensible  for  one  or  two  revolutions  of  Venus  or  Mars), 
other  as  «een      Continue  observations  on  the  same  planet,  until  it  comes 
from  the  sun.  into  the  ecliptic  the  second  time  after  the  first  observation, 
or  to  the  same  node  again;  and  the  time  elapsed,  is  the  time  of 
a  revolution  of  that  planet  round  the  sun.     From  such  observa- 
tions the  periodical  time  of  Venus   became  well  known  to 
astronomers,  long  before  they  had  opportunities  to  decide  it 
by  comparing  its  transits  across  the  sun's  disc ;  and  by  thus 
knowing  its  periodical  time  and  motion,  they  were  enabled  to 
calculate  the  times  and  circumstances  of  the  transits  which 
happened  in  1761,  and  in  1769;  save  those  resulting  from 
parallax  alone. 

First  idea  of  (109.)  On  comparing  the  time  that  a  planet  remains  on 
of'the^piTn*  eac^  s^e  °f tne  ecliptic*  we  can  form  some  idea  of  the  position 
»ts.  of  its  apogee  and  perigee.  If  it  is  observed  to  be  on  each  side 

of  the  ecliptic  the  same  length  of  time,  then  it  is  evident  that 
the  orbit  of  the  planet  is  circular,  or  that  its  longer  axis  coin- 
cides with  its  nodes.  If  it  is  observed  to  be  a  shorter  time 
north  of  the  plane  of  the  ecliptic  than  south  of  it,  then  it  is 
evident  that  its  perigee  is  north  of  the  ecliptic;  but  nothing 
more  definite  can  be  drawn  from  this  circumstance. 
Final  remits.  (110.)  Finally.  By  the  combination  of  the  different 
methods,  explained  in  articles  (98  ),  (  100 ),  (  101 ),  ( 105  ), 
(107  ),  and  (108),  and  extending  the  observations  through 
a  lo  i<r  course  of  years,  and  from  age  to  age,  the  times  of  rev- 
olution, the  mean  relative  distances  of  the  planets  from  the 
sun,  were  approximated  to,  step  by  step,  until  a  great  degree 
of  exactness  was  attained,  and  the  following  were  the  results : 

Sidereal  Revolution.        Mean  distance  from  Q 

Mercury,-  -  -     87.969258  0.387098 

Venus,      -  -  -  224.700787  0.723332 

Earth,      -  -  -  365.256383  1.000000 

Mars,       -  -  -  686.979646  1.523692 

Jupiter,    -  -  4332.584821  5.202776 

Saturn,     -  -  10759.219817  9.538786 

Uranus,    -  -  30686.820830  19.182390 


PLANETARY    MOTION.  727 

(  111.)  By  inspecting  the  preceding  table,  we  find  that  the  CHAP.  via. 
greater  the  distance  from  the  sun,  the  greater  the  time  of  Tim 


revolution  ;  but  the  ratio  for  the  time  is  greater  than  the  ratio  olution   aud 

1,1  distances 

corresponding  to  distance  ;  yet  we  cannot  doubt  that  some  Cemim«Mi 
connection  exists  between  these  ratios. 

For  instance,  let  us  compare  the  Earth  with  Jupiter.  The 
ratio  between  their  times  of  revolution,  is  near  12. 

The  ratio  between  their  relative  distances  from  the  sun,  as 
we  perceive,  is  nearly  5.2. 

The  square  of  12  is  144  ;  the  cube  of  5.2  is  near  141. 
But  12  is  a  little  greater  than  the  real  ratio  between  the 
times  of  revolution,  and  5.2  is  not  quite  large  enough  for  the 
ratio  of  distance;  and  by  taking  the  correct  ratios,  they  seem 
to  bear  the  relation  of  square  to  cube. 

Without  a  very  rigid  or  <jlose  examination,  we  perceive 
that  five  revolutions  of  Jupiter  are  nearly  equal  to  two  revolu- 
tions of  Saturn;  that  is,  f-  is  nearly  the  ratio  between  their 
times  of  revolution. 

By  inspecting  the  column  of  distances,  we  perceive  that 
the  ratio  of  the  distances  of  these  two  planets,  is  nearly  f  f  ; 
and  if  we  square  the  first  ratio,  and  cube  the  second,  we  shall 
have  nearly  the  same  ratio. 

Now  let  us  compare  two  other  planets,  say    Venus  and  Result    d>» 
Mars,  more  exactly.  cowed. 

Their  ratio  of  revolution  is       686.979  log.    -  2.836948 

224.701  log-   -  2.351601 

Log.  of  the  ratio,                     ^         -  0.485347 

Multiply  by                                    -  2 

Log.  of  the  square  of  the  ratio  of  time,  0.870694 

Their  ratio  of  distance  is,         15.23692  log.  -  1.182883 

"7^23332  log.  -    859323 

Log.  of  the  ratio,          -  -         0.323560 

Multiply  by    -  .        .  3 

Log.  of  the  cube  of  the  ratio  of  distance,  0.970680 
Thus  we  perceive  that  the  squares  of  the  times  of  revolu- 
tion. are  to  each  other  as  the  cubes  of  the  mean  distances  of 


128  ASTRONOMY 

CHAP,  vin.  the  planets  from  the  sun,*  and  this  is  called  Kepler's  third 
Kepler's  law  :  and  it  was  by  such  numerical  comparisons  that  Kepler 
discovered  the  law.f 

We  may  now  recapitulate  the  three  laws  of  the  solar  sys- 
tem, called  Kepler's  laws,  as  they  were  discovered  by  that 
philosopher. 

1st.  The  orbits  of  the  planets  are  ellipses,  of  which  the  sun 
occupies  one  of  the  foci. 

^d.  The  radius  vector  in  each  case  describes  areas  about  the 
focus,  which  are  proportional  to  the  times. 

3c?.  The  squares  of  the  times  of  revolution  are  to  each  other 
as  the  cubes  of  the  mean  distances  from  the  sun. 

*  For  a  concise  mathematical  view  of  this  subject,  we  give 
the  following:  Let  d  and  D  represent  mean  distances  from 
the  sun,  and  t  and  T  the  times  of  revolution.  Then 

T  D 

~=  n,     ~j  —  m>    n  an(J  m  taken   to  represent  the  ratios. 

Square  the  1st  equation  and  cube  the  2d.     Then 

T2  D* 

~=n2,    and     ~^=m\ 

But  by  inspection  we  know  that 

n2=m3;  therefore,  — =  --1,  or,  t2  :  T2  ::d*  :  D3. 
t2         a3 


f  It  appears  that  Kepler  did  not  compare  ratios,  as  we  have  done  ; 
but  took  the  more  ponderous  method  of  comparing  the  elements  of  the 
ratios  (the  numbers  themselves )  ;  for,  says  the  historian  :  —  It  was  on 
the  8th  of  March,  1618,  that  it  first  came  into  Kepler's  mind  to  com- 
pare the  powers  of  the  numbers  which  express  their  revolutions  and 
distances  ;  and  by  chance  he  compared  the  squares  of  the  times  with 
the  cubes  of  the  distances ;  but  from  too  great  anxiety  and  impa- 
tience, he  made  such  errors  in  computation,  that  he  rejected  the  hy- 
pothesis as  false  and  useless ;  but  on  examining  almost  every  other 
relation  in  vain,  he  returned  to  the  same  hypothesis,  and  on  the  15th 
of  May,  of  the  same  year,  he  renewed  his  calculation  with  complete 
success,  and  established  this  law,  which  has  rendered  his  name  im- 
mortal 


SOLAR   PARALLAX. 


CHAPTER     IX. 

TRANSITS    OF    VENUS    AND    MERCURY. HOW    SUN's    HORIZONTAL 

PARALLAX    DEDUCED 

( 112. )  WE  have  thus  far  been  very  patient  in  our  inves-    CHAP,  ix 
tigations  —  groping  along  —  finding  the  form  of  the  planetary  Attem^  l« 
orbits,  and  their  relative  magnitudes;  but,  as  yet,  we  know  find  the  iun'» 
nothing  of  the  distance  to  the  sun ;  save  the  indefinite  fact,  Parallax- 
that  it  must  be  very  great,  and  its  magnitude  great;  but 
how  great  we  can  never  know,  without  the  sun's  parallax. 
Hence,  to  obtain  this  element,  has  always  been  an  interesting 
problem  to  astronomers. 

The  ancient  astronomers  had   no   instruments  sufficiently    Dim^iti., 
refined  to  determine  this  parallax  by  direct  observation,  in  the  of    ancient 
manner  of  finding  that  of  the  moon  (Art.  60),  and  hence  the  Mtronoineiri- 
ingenuity  of  men  was  called  into  exercise  to  find  some  artifice 
to  obtain  the  desired  result. 

After  Kepler's  laws  were  established,  arid  the  relative  dis- 
tances of  the  planets  made  known,  it  was  apparent  that  their 
real  distance  could  be  deduced,  provided  the  distance  between 
the  earth  and  any  planet  could  be  made  known. 

(113.)  The  relative  distances  of  the  earth  and  Mars,  from   paranajKflf 
the  sun  (as  determined  by  Kepler's  law)  are  as  1  to  1.5237 ;  Mar». 
and  hence  it  follows  that  Mars,  in  its  oppositions  to  the  sun, 
is  but  about  one  half  as  far  from  the  earth  as  the  sun  is;  and 
therefore  its  parallax  (Art.  60)  must  be  about  double  that 
of  the  sun ;  and  several  partially  successful  attempts  were 
made  to  obtain  it  by  observation. 

On  the  15th  of  August,  1719,  Mars  being  very  near  its 
opposition  to  the  sun,  and  very  near  a  star  of  the  5th  mag- 
nitude,  its  parallax  became  sensible ;  and  Mr.  Maraldi,  an  tion  u>  u* 
Italian  astronomer,  pronounced  it  to  be  27".  The  relative 
distance  of  Mars,  at  that  time,  was  1.37,  as  determined  from 
its  position  and  the  eccentricity  of  its  orbit. 

But  horizontal  parallax  is  the  angle  under  which  the  earth 
appears ;  and,  at  a  greater  distance,  it  will  appear  under  a 
9 


130  ASTRONOMY. 

CHAP.  ix.  less  angle.  The  distance  of  Mars  from  the  earth,  at  that 
time,  was  .37,  and  the  distance  of  the  sun  was  1 ;  therefore, 
1  :  .37  ::  27"  :  9".99,  or  10",  nearly,  for  the  sun's  horizon- 
tal parallax. 

On  the  6th  of  October,  1751,  Mars  was  attentively  ob- 
serve<^  ^J  Wargentin  and  Lacaille  (it  being  near  its  opposi- 

Lacaiiie  tion  to  the  sun),  and  they  found  its  parallax  to  be  24" .6, 
from  which  they  deduced  the  mean  parallax  of  the  sun,  10".7. 
But  at  that  time,  if  not  at  present,  the  parallax  of  Mars 
could  not  be  observed  directly,  with  sufficient  accuracy  to 
satisfy  astronomers ;  for  no  observer  could  rely  on  an  angu- 
lar measure  within  2" ;  for  full  that  space  was  eclipsed  by 
the  micrometer  wire. 

Dr.  Hal-  (114.)  Not  being  satisfied  with  these  results,  Dr.  Halley, 
'*  ^ss68'  an  English  astronomer,  very  happily  conceived  the  idea  of 
finding  the  sun's  parallax  by  the  comparisons  of  observa- 
tions made  from  different  parts  of  the  earth,  on  a  transit  of 
Venus  over  the  sun's  disc.  If  the  plane  of  the  orbit  of  Venus 
coincided  with  the  orbit  of  the  earth,  then  Venus  would  come 
between  the  earth  and  sun,  at  every  inferior  conjunction,  at 
intervals  of  584.04  days.  But  the  orbit  of  Venus  is  inclined 
to  the  orbit  of  the  earth  by  an  angle  of  3°  23'  28" ;  and,  in 
the  year  1800,  the  planet  crossed  the  ecliptic  from  south  to 
north,  in  longitude  74°  54'  12",  and  from  north  to  south,  in 
longitude  254°  54'  12":  the  first  mentioned  point  is  called 
The  nodes  the  ascending  node  ;  the  last,  the  descending  node.  The  nodes 

of  Venus.     retrograde  3r  10»  in  a  century. 

Whattimei      (115.)  The  mean  synodical  revolution  of  584  days  corre- 
i*  the  year  gp0nds  wjfc}1  no  a]iquot  part  of  a  year ;  and  therefore,  in  the 

transits  may     *  .     *         / 

take  place,    course  of  time,  these  conjunctions  will  happen  at  different 

points  along  the  ecliptic.  The  sun  is  in  that  part  of  the  ecliptic 

near  the  nodes  of  Vnnus,  June  5th  and  December  6th  or  7th ; 

and  the  two  last  transits  happened  in  1761  and  in  1769;  and 

from  these  periods  we  date  our  knowledge  of  the  solar  parallax. 

Revoin-      (  116.)  The  periodical  revolution  of  the  earth  is  365.256383 

•tio,»    com.  dayg  and  that  of  yermg  ig  224.700787 ;  and  as  numbers  they 

are  nearly  in  proportion  of  13  to  8. 

From  this  it  follows,  that  eight  revolutions  of  the  earth 


SOLAR    PARALLAX.  131 

require  nearly  the  same  time  as  13  revolutions  of  Venus;    CHAP. ex 
and,  of  course,  whenever  a  conjunction   takes  place,  eight 
years  afterward  another  conjunction  will  take  place  very  near 
the  same  point  in  the  ecliptic.* 


*  The  ratio  of  the  times  of  these  revolutions  is  directly     Compara. 

224.700787  tive  motions 

compared,  as  terms  of  a  fraction,  thus,  TTTTF-TFTTTTTTT;  and  it  is  ofVennsanu 

o05.25uo81  the  earth. 

manifest  that  365.256383  days,  multiplied  by  the  number 
224700787,  will  give  the  same  product  as  224  700787  days 
multiplied  by  the  number  365256383 ;  that  is,  after  an  elapse 
of  224700787  years,  the  conjunction  will  take  place  at  the 
same  point  in  the  heavens;  and  all  intermediate  conjunctions 
will  be  but  approximations  to  the  same  point :  and  to  obtain 
these  approximate  intervals,, we  reduce  the  above  fraction  to 
its  approximating  fractions,  by  the  principle  of  continued 

fractions.       (  See  Robinson's  Arithmetic.  ) 

The  approximating  fractions  are 

1        1       2       3        8        235 
I'       2'       3'       5'       13'       382* 

To  say  nothing  of  the  first  two  terms,  these  fractions  show 
that  two  revolutions  of  the  earth  are  near,  in  length  of  time, 
to  three  revolutions  of  Venus ;  three  revolutions  of  the  earth 
a  nearer  value  to  five  revolutions  of  Venus ;  and  eight  revo- 
lutions of  the  earth  a  still  nearer  value  to  13  revolutions  of 
Venus ;  and  235  revolutions  of  the  earth  a  very  near  value 
to  382  revolutions  of  Venus. 

The  period  of  eight  years,  under  favorable  circumstances, 
will  bring  a  second  transit  at  the  same  node :  but  if  not  in 
eight  years,  it  will  be  235  years,  or  235+8=243  years. 

For  a  transit  at  the  other  node,  we  must  take  a  period  of 
235 — 8  years,  divided  by  2,  or  113  years;  and  sometimes 
the  period  will  be  eight  years  less  than  this,  or  105  years 
The  first  transit  known  to  have  been  observed  was  in  1639, 
December  4th ;  to  this  add  235  years,  and  we  have  the  time 
of  the  next  transit,  at  the  same  node,  1874,  December  8th; 
and  eight  years  after  that  will  be  another,  1882,  December 
6th.  The  first  transit  observed  at  the  ascending  node,  was 


132  ASTRONOMY. 

CHAP.  ix.       If  the  proportion  had  been  exactly  as  13  to  8,  then  the 

Periods  of  conjunctions  would  always  take  place  exactly  at  the  same 

conjunction.       -^  .  kut  as  jt  jg  faQ  points  of  conjunction  in  the  heavens 

at   the   same  r  .  J 

time  of  the  are  east  and  west  of  a  given  point,  and  approximate  nearer 
ye&r-  and  nearer  to   that   point  as  the   periods  are  greater   and 

greater. 

only  two      To  be  more  practical,  however,  the  intervals  between  con- 
*"   °"  junctions  are  such,  combined  with  a  slight  motion  of  the  nodes, 
tervals  of  8  that  the  geocentric  latitude  of  Venus,  at  inferior  conjunctions 
years.          near  ^]ie  ascending  node,  changes  about  19'  30"  to  the  north, 
in  the  period  of  about  eight  years.     At  the  descending  node, 
it  changes  about  the  same  quantity  to  the  southward,  in  the 
same  period  ;  and  as  the  disc  of  the  sun  is  but  little  over  32', 
it  is  impossible  that  a  third  transit  should  happen  16  years 
after  the  first;  hence  only  two  transits  can  happen,  at  the 
same  node,  separated  by  the  short  interval  of  eight  years. 
Periods  be.       (117.)  If  at  any  transit  we  suppose  Venus  to  pass  directly 
^e  center  of  the  sun,  as  seen  from  the  center  of  the 
earth  —  that  is,  pass  conjunction  and  node  at  the  same  time  — 
at  the   end  of  another  period  of  about  eight  years,  Venus 
would  be  19'  30"  north  or  south  of  the  sun's  center;  but  as 
the  semidiameter  of  the  sun  is  but  about  16',  no  transit  could 
happen  in  such  a  case  ;  and  there  would  be  but  one  transit 
at  that  node  until  after  the  expiration  of  a  long  period  of  235 
or  243  years. 

After  passing  the  period  of  eight  years,  we  take  a  lapse  of 
105  or  113  years,  or  thereabouts,  to  look  for  a  transit  at  the 
other  node. 

Transits  ^  ^g  ^  Knowing  the  relative  distances  of  Venus,  and  the 
pnted.  earth,  from  the  sun  —  the  positions  and  eccentricities  of  both 
Dr.  Haiiey  orbits  —  also  their  angular  motions  and  periodical  revolutions  — 
io°find  the  everv  circumstance  attending  a  transit,  as  seen  from  the 
sun's  parai-  earth's  center,  can  be  calculated;  and  Dr.  Halley,  in  1677, 
lax*  read  a  paper  before  the  London  Astronomical  Society,  in 


jrext  note  in  ly^l,  june  5tn  ;  eight  years  after,  1769,  June  3d,  there 
was  another  ;  and  the  next  that  will  occur,  at  that  node,  will 
be  in  2004,  June  7th,  235  years  after  1769. 


SOLAR    PARALLAX.  133 

which  he  explained  the  manner  of  deducing  the  parallax  of    CHAP,  1X» 
the  sun,  from  observations  taken  on  a   transit  of  Venus  or 
Mercury  across  the  sun's  disc,  compared  with  computations 
made  for  the   earth's  center,  or  by  comparing  observations 
made  on  the  earth  at  great  distances  from  each  other. 

The  transits  of  Venus  are  much  better,  for  this  purpose,       why  the 
than  those  of  Mercury ;  as  Venus  is  larger,  and  nearer  the  transits    of 

J  '  m  Venus       are 

earth,  and  its  parallax  at  such  times  much  greater  than  that  better  adapt- 
of  Mercury ;  and  so  important  did  it  appear,  to  the  learned  ed   to  «iv9 
world,   to  have   correct  observations  on  the  last  transit  of  raj^  "thwi 
Venus,  in  1769,  at  remote  stations,  that  the  British,  French,  those  of  Mer- 
and  Russian  governments  were  induced  to  send  out  expedi-  cnry* 
tions  to  various  parts  of  the  globe,  to  observe  it.     "  The  fa- 
mous expedition  of  Captain  Cook,  to  Otaheite,  was  one  of 
them." 

(119.)  The  mean  result  of  all  the  observations  made  on  The  result 
that  memorable  occasion,  gave  the  sun's  parallax,  on  the  day 
of  the  transit  (3d  of  June),  8".5776.  The  horizontal  paral- 
lax, at  mean  distance,  may  be  taken  at  8". 6 ;  which  places 
the  sun,  at  its  mean  distance,  no  less  than  23984  times  the 
length  of  the  earth's  semidiameter,  or  about  95  millions  of 
miles. 

This  problem  of  the  sun's  horizontal  parallax,  as  deduced    The  hnpor. 
from  observations  on  a  transit  of  Venus,  we  regard  as  the  tance  of  tbu 
most  important,  for  a  student  to  understand,  of  any  in  astro-  Pr 
nomy ;  for  without  it,  the  dimensions  of  the  solar  system,  and 
the  magnitudes  of  the  heavenly  bodies,  must  be  taken  wholly 
on  trust;  and  we  have  often  protested  against  mere  facts 
being  taken  for  knowledge. 

( 120.)  We  shall  now  attempt  to  explain  this  whole  matter    A  general 
on  general  principles,  avoiding  all  the  little  minutiae  which  exPIanatioa 
render  the  subject  intricate  and  tedious ;  for  our  only  object 
is  to  give  a  clear  idea  of  the  nature  and  philosophy  of  the 
problem. 

Let  S  (Fig.  26)  represent  the  sun,  and  m  n  and  P  Q  small 
portions  of  the  orbits  of  Venus  ftnd  the  earth. 

As  these  two  bodies  move  the  same  way,  and  nearly  in  the 
same  plane,  we  may  suppose  the  earth  stationary,  and  Venus 


134 


ASTRONOMY 


At  abstract 
proposition 
for    the  pur- 
pose of  illus- 
tration. 


to  move  with  an  angular  velocity 
equal  to  the  difference  of  the  two 

When  the  planet  arrives  at  v,  an 
observer  at  A  would  see  the  planet 
projected  on  the  sun,  making  a  dent 
at  v'. 

But  an  observer  at  G  would  not 
see  the  same  thing  until  after  the 
planet  had  passed  over  the  small  aie 
v  g,  with  a  velocity  equal  to  the  dit'- 
erence  between  the  angular  motion 
of  the  two  bodies;  and  as  this  will 
require  quite  an  interval  of  absolute 
time,  it  can  be  detected ;  and  it  mea- 
sures the  angle  A  v'  G;  an  angle 
under  which  a  definite  portion  of  the 
earth  appears  as  seen  from  the  sun. 

(121.)  To  have  a  more  definite 
idea  of  the  practicability  of  this  me- 
thod, let  us  suppose  the  parallactic 
angle,  A  v'  G,  equal  to  10",  and  in- 
quire how  long  Venus  would  be  in 
passing  the  relative  arc  v  q. 

Venus,  at  its  mean  rate,  passes     -     1°  36'  8"  in  a  day. 

The  earth,  «  "  59' 8"      " 

The  relative,  or  excess  motion  of  Venus  for  a  mean  solar 
day  is  then  37'. 

Now,  as  37'  is  to  24h.  so  is  10"  to  a  fourth  term;  or,  as 
2-220"  :  1440m.  ::  10"  :  6  m.  29  s. 

Now  if  observation  gave  more  than  6  minutes  and  29  sec- 
onds, we  shall  conclude  that  the  parallactic  angle  was  more 
fchan  10";  if  less,  less.  But  this  is  an  abstract  proposition. 
When  treating  of  an  actual  case  in  place  of  the  mean  motion, 
We  must  take  the  actual  angular  motions  of  the  earth  and 
Venus  at  that  time,  and  we'must  know  the  actual  position  of 
the  observers  A  and  G  in  respect  to  each  other,  and  the  po- 
sition of  each  in  relation  to  a  line  joining  the  center  of  the 


SOLAR    PARALLAX.  135 

earth  and  the  center  of  the  sun  ;  and  then  by  comparing  the  CHAP.  IX 
local  time  of  observation  made  at  A,  with  the  time  at  G,  and 
referring  both  to  one  and  the  same  meridian,  we  shall  have  the 
interval  of  time  occupied  by  the  planet  in  passing  from  v  to 
q,  from  which  we  deduce  the  parallactic  angle  A  v'  G,  and 
from  thence  the  horizontal  parallax. 

The  same  observations  can  be  made  when  the  planet  passes 
off  the  sun,  and  a  great  many  stations  can  be  compared  with 
A,  as  well  as  the  station  G.  In  this  way,  the  mean  result  of 
a  great  many  stations  was  found  in  1761,  and  in  1769,  and 
the  mean  of  all  cannot  materially  differ  from  the  truth. 

(  122.)  There  is  another  method  of  considering  this  whole  Another  me- 
subject,  which  is  in  some  respects  more  simple  and  preferable  thodo[deda- 

J  cing  the  pro- 

to  the  one  just  explained.  It  is  for  the  observers  at  every  biem 
station  to  keep  the  track  of  the  transit  all  the  way  across  the 
sun's  disc,  and  take  every  precaution  to  measure  the  length 
of  chord  upon  the  disc,  which  can  be  done  by  carefully  noting 
the  times  of  external  and  internal  contacts,  and  the  begin- 
ning and  end  of  the  transit,  and  at  short  intervals  carefully 
measuring  the  distance  of  the  planet  to  the  nearest  edge  of 
the  sun  by  a  micrometer. 

If  the  parallax  is  sensible,  it  is  evident  that  two  observers,   Situation  of 
situated  in  different  hemispheres,  will  not  obtain  the  same 
chord.     For  example,  an  observer  in  the  northern  hemisphere, 
as  in  Sweden  or  Norway,  will  see  Venus  traversing  a  more 
southern  chord  than  an  observer  in  the  southern  hemisphere. 

Now  if  each  observer  gives  us  the  length  of  the  chord  as  ob- 
served by  himself,  and,  knowing  the  angular  diameter  of  the 
sun,  we  can  compute  the  distance  of  each  chord  from  the 
sun's  center,  and  of  course  we  then  have  the  angular  breadth 
of  the  zone  on  the  sun's  disc  between  them.  But  as  this 
zone  is  formed  by  straight  lines  passing  through  the  same 
point,  the  center  of  Venus,  its  absolute  breadth  will  depend  on 
its  distance  from  the  point  v;  that  is,  the  two  triangles  ABv 
and  a  b  v  (  Fig.  27)  will  be  proportional,  and  we  have 

Av:av::A£:ab. 

But  the  first  three*  of  these  terms  are  known  ;  therefore  the 
fourth,  a  b,  is  known  also  ;  and  if  any  definite  angular  space 


13G 


ASTRONOMY. 


CHAP.  IX. 


Under  what 

circumstan- 
ces this  me- 
thod should 
not  be  used. 


Transits  Oil 
Mercury  not 
important. 


Revolutions 
of  Mercury 
and  the  earth 
compared. 


Fig.  27.  on  the  sun  becomes  known,  the  whole  sem- 
idiameter  becomes  known,  and  from  thence 
the  horizontal  parallax  is  immediately  dedu- 
ced* 

(123.)  The  accuracy  of  this  method  should  bd 
questioned  when  Venus  passes  near  the  sun'» 
center,  for  the  two  chords  are  never  more  than 
30"  asunder,  and  hence  they  will  not  percepti- 
bly differ  in  length  when  passing  near  the  sun'? 
center,  and  Venus  will  be  upon  the  sun  nearly 
the  same  length  of  time  to  all  observers. 

( 124.)  The  apparent  diameter  of  Mercury 
and  Venus  can  be  very  accurately  measured 
when  passing  the  sun's  disc.  In  1769  the  di- 
ameter of  Venus  was  observed  to  be  59". 

( 125.)  The  same  general  principles  applj- 
to  the  transits  of  Mercury  and  Venus ;  but  those 
of  Mercury  are  not  important,  on  account  of  the 
smaller  parallax  and  smaller  size  of  that  planet : 
but  owing  to  the  more  rapid  revolution  of  Mer- 
cury, its  transits  occur  more  frequently.  The 
frequent  appearance  of  this  planet  on  the  face 
of  the  sun,  gives  to  astronomers  fine  opportu- 
nities to  determine  the  position  of  its  node  and 
the  inclination  of  its  orbit. 
In  1779,  M.  Delambre,  from  observations  on  the  transit  of 
May  7,  placed  the  ascending  node,  as  seen  from  the  sun,  in 
longitude  45°  57'  3".  From  the  transit  of  the  8th  of  May, 
1845,  as  observed  at  Cincinnati,  it  must  have  been  in  longi- 
tude 46°  31'  10";  this  gives  it  a  progressive  motion  of  about 
1°  10'  in  a  century.  The  inclination  of  the  orbit  is  7°  0'  13". 
The  periodical  time  of  revolution  is  87.96925  days;  that  of 
the  earth  is  365.25638  days,  and  by  making  a  fraction  of 
these  numbers,  and  reducing  as  in  the  last  text  note,  we  find 


•  That  is,  as  the  real  diameter  of  the  sun,  is  to  the  real  diameter  of 
the  earth,  so  is  the  sun's  angular  semidiameter  to  its  horizontal  par- 
allax. ( See  66). 


PLANETARY  PARALLAX.  137 

that  6,  7,  13,  33,  46,  79,  and  520  years,  or  revolutions  of  the  CHAP,  ix. 
earth  nearly  correspond  to  complete  revolutions  of  Mercury. 
Hence  we  may  look  for  a  transit  in  6,  7,  13,  33,  46,  &c., 
years,  or  at  the  expiration  of  any  combination  of  these  years 
after  any  transit  has  been  observed  to  take  place ;  and  by 
examining  the  following  table,  the  years  will  be  found  to  fol-  [j,*1*** 
low  each  other  by  some  combination  of  these  numbers.  8j», 

The  following  is  a  list  of  all  the  transits  of  Mercury  that 
have  occurred,  or  will  occur,  between  the  years  1800  an<* 
1900: 

At  the  ascending  node.  At  the  descending  node. 

May  7. 

-  May  5. 
May  8. 

-  May  6. 
May  9. 


1802, 

-  -  -  Nov.  8. 

1799,  -  - 

1822, 

-  -  -  Nov.  4. 

1832,  -  - 

1835, 

-  -  -  Nov.  7. 

1845,  -  - 

1848, 

-  -  -  Nov.  9.  , 

1878,  -  - 

1861, 

-  -  Nov.  11.  ' 

1891,  -  - 

1868, 

-  -  -  Nov.  4. 

1881, 

-  -  -  Nov.  7. 

1894, 

-  -  -  Nov.  10. 

CHAPTER   X. 

THE   HORIZONTAL   PARALLAXES    OF  THE  PLANETS    COMPUTED,  AND 
FROM    THENCE    THEIR    REAL    DIAMETERS   AND    MAGNITDDES. 

( 126.)  HAVING  found  the  real  distance  to  the  sun,  and  the    CHAP.  X. 
sun's  horizontal  parallax,  we  have  now  sufficient  data  to  find     Real  mafr 
the  real  distance,  diameter,  and  magnitude,  of  every  planet  nitudes  and 

.      A.  .  distances cat 

in  the  solar  system.  now  be  de. 

In  Art.  60  we  have  explained,  or  rather  defined,  the  hori- 
zontal  parallax  of  any  body  to  be  the  angle  under  which  the 
semidiameter  of  the  earth  appears,  as  seen  from  that  body ; 
and  if  the  earth  were  as  large  as  the  body,  the  semi-diame- 
ter of  the  body,  and  its  liorizontal  parallax,  would  have  the 
same  value.  And,  in  general,  the  diameter  of  the  earth  is  to 
the  diameter  of  any  other  planetary  body,  as  the  horizontal 
parallax  of  that  body  is  to  its  apparent  semidiametev. 

The  mean  horizontal  parallax  of  the  sun,  as  determined  in 


J38  ASTRONOMY. 

CHAP  x.    the  last  chapter,  is  8".6;  the  semidiaraeter  of  the  sun,  at  th€ 
Re7i"<iia.  corresponding  mean  distance,  is  16'  1",  or  961".     Now  let  d 
meter  of  the  represent  the  real  diameter  of  the  earth,  and  D  that  of  the 
mined   "  ""  sun»  tnen  we  saa^  nave  the  following  proportion : 

d    :     D     ::     8".6     :     961".0. 

But  d  is  7912  miles;  and  the  ratio  of  the  last  two  terms  is 
111.74;  therefore  Z>=(111.74)(7912)=884087  miles. 
Real  dis-      ( 127.)  The  sun's  horizontal  parallax  is  the  angle  at  the 
tance      be-  base  Of  a  right,  angled  triangle;  and  the  side  opposite  to  it  is 

tween       the  ~,  . 

earth  and  sun  •**  radius  ot  the  earth  (which,  tor  the  sake  ot  convenience, 
determined,   we  now  call  unity).     Let  x  represent  the  radius  of  the  earth's 
orbit;  then,  by  trigonometry, 

sin.  8".6     :     1     :  :     sin.  90°     :     x\ 

cin    Q0° 

Therefore,  *=^pL==log.  10.00000— log.  5.620073  * 

That  is,  the  log.  of  3=4.379927,  or  a?=23984 ;  which  is 
the  distance  between  the  earth  and  sun,  when  the  semidia- 
meter  of  the  earth  is  taken  for  the  unit  of  measure ;  but,  for 
general  reference,  and  to  aid  the  memory,  we  may  say  the 
distance  is  24000  times  the  earth's  semidiameter. 

(128.)  Now  let  us  change  the  unit  from  the  semidiameter 
of  the  earth  to  an  English  mile ;  and  then  the  distance  be- 
tween the  earth  and  sun  is 
Distance  i«  (3956)(23984)=94880706 ; 

f*und     num- 

l^eif  and,  in  round  numbers,  we  say  95  millions  of  miles. 

By  Kepler's  third  law,  we  know  th,e  relative  distances  of 

*  Students  generally  would  be  unable  to  find  the  sine  of  8". 6,  or  the 
sine  of  any  other  very  small  arc  ;  for  the  directions  given  in  common 
works  of  trigonometry  are  too  gross,  and,  indeed,  inaccurate,  to  meet 
the  demands  of  astronomy. 

On  the  principle  that  the  sines  of  small  arcs  vary  as  the  arcs  them- 
selves, we  can  find  the  sine  of  any  small  arc  as  follows : 

Sine  of  1',  taken  from  the  tables,  is     -         -         -         -     6.463726 
Divide  by  60,  that  is,  subtract  the  log.  of  60,  -         -          1.778151 

The  sine-of  1",  therefore,  is 4.  685575 

Multiply  by  the  number  8.6  ;  that  is,  add  log.         -          0.  934498 

Tim  nine  of  8".6,  therefore,  must  be,  -         -         -         -     5.  620073 
In  the  same  manner,  find  the  sine  of  any  other  small  are. 


PLANETARY  PARALLAX.  139 

all  tiie  planets  from  the  sun ;  and  now,  having  found  the  real    Cm?,  x. 
distance  of  the  earth,  we  may  have  the  distance  in  miles,  by       HOW   to 
multiplying  the  distance  of  the  earth  by  the  ratio  correspond-  find  the  dl* 
ing  to  any  other  planet.     Thus,  for  the  distance  of  Venus,  planet   from 
we    multiply    94880706    by    .72333 ;    and    the   result    is  the    su»   in 
68629960  miles,  for  the  distance  of  Venus:  and  proceed,  in 
the  same  manner,  for  the  distance  of  any  other  planet. 

(129.)  By  observations  taken  on  the  transit  of  Venus,  in     TO  find  the 
1769,  it  was  concluded  that  the  horizontal  parallax  of  that  y*™ne,ter  °f 
planet  was  30".4;   and  its  semidiameter,  at  the  same  time, 
was  29".2.     Hence  (Art.  126),    304  :  292  :  :  7912  :    to  a 
fourth  term;    which  gives   7599  miles  for  the  diameter  of 
Venus. 

(130.)  ^Ye  cannot  observe  the  horizontal  parallax  of  Ju- 
piter,  Saturn,  or  any  other  very  remote  planet:  if  known  at 
all,  it  becomes  known  by  computation ;  but  the  parallax  can  »erved. 
be  known,  when  the  real  distance  is  known;  and,  by  Kepler's 
third  law,  and  the  solar  parallax,  we  do  know  all  the  planetary 
distances ;  and  can,  of  course,  compute  any  particular  hori- 
zontal parallax. 

For  the  horizontal  parallax  of  Jupiter,  when  at  a  distance 
from  the  earth  equal  to  its  mean  distance  from  the  sun,  we 
proceed  as  follows : 

The  parallax,  or  the  semidiameter  of  the  earth,  when  seen 
at  the  distance  of  the  sun,  is  8".6.  When  seen  from  a  greater 
distance,  the  angle  would  be  proportionally  less. 

Put  k  equal  to  the  horizontal  parallax  of  Jupiter  ;  then  we 

have,  -    5.202776  :  1  ::  8".6  :  k;     or    A=T|.'£_. 

From  this,  we  perceive,  that  if  we  divide  the  sun's  horizontal       HOW  u> 
parallax  by  the  ratio  of  a  planet's  distance  from  the  sun,  the  comPute  fh« 
quotient  will  be  the  horizontal  parallax  of  the  planet,  when  at  a  the  planet. 
distance  from  the  earth  equal  to  its  mean  distance  from  the  sun. 

(131.)  To  find  the  diameter  of  a  planet,  in  relation  to  the       HOW   to 
diameter  of  the  earth,  we  have  a  similar  proportion  as  in  Art.  find  the  real 

i  •>£»  i  /.-ITT  •  diameters  of 

l'2b :  and  to  find  the  diameter  of  Jupiter,  we   proceed  as  the  piaMU. 
follows : 

The  greatest  apparent  diameter  of  Jupiter,  as  seen  from 


140  ASTRONOMY. 

CHAP.  x.  the  earth,  is  44".5;  the  least  is  30".  1;  therefore  the  mean, 
as  seen  from  the  sun.  cannot  be  far  from  37  ".3,  and  the  semi- 
diameter  18".  65;  La  Place  says  it  is  18".  35;  and  this  value 
we  shall  use.  Now,  as  in  Art.  126,  let  d=7912,  D=  the 

O//   /"» 

unknown  diameter  of  Jupiter  ;       OAOT-T/?     is    its    horizontal 


parallax,  and  18".35  its  corresponding  semidiameter  ;  then,  as 
in  Art  126.          7912.     :     D    :  :     -  :     18.85; 

Therefore  p= 

87900  miles. 

In  the  same  manner,  we  may  find  the  diameter  of  any 
other  planet. 

Jupiter  not  We  have  just  seen  that  the  diameter  of  Jupiter  is  11.11 
spherical.  times  the  diameter  of  the  earth  ;  but  this  is  the  equatorial 
diameter  of  the  planet.  Its  polar  diameter  is  less,  in  the 
proportion  of  167  to  177,  as  determined  by  the  mean  of  many 
micrometrical  measurements  ;  which  proportion  gives  82930 
miles,  for  the  polar  diameter  of  Jupiter.  These  extremes 
give  the  mean  diameter  of  Jupiter,  to  the  mean  diameter  of 
the  earth,  as  10.8  to  1. 

HOW  to  find      (132.)  But  the   magnitudes  of  similar  bodies  are  to  one 
the    mag»i-  anot|)er  as  fae  Cll]jes  Of  their  like  dimensions  ;  therefore  the 

tude    of  i  >e 

planets.  magnitude  of  Jupiter  is  to  that  of  the  earth,  as  (10.8)3  to 
1,  and  from  thence  we  learn  that  Jupiter  is  1260  times 
greater  than  the  earth. 

In  the  same  manner  we  may  find  the  magnitude  of  any 
other  planet,  and  it  is  thus  that  their  magnitudes  have  often 
been  determined,  and  the  results  may  be  seen  in  a  concise 
form  in  Table  III,  which  gives  a  summary  view  of  the  solar 
system. 

The  masses  and  attractions  of  the  different  planets  will  be 
investigated  in  physical  astronomy,  after  we  become  acquain- 
ted with  the  theory  of  universal  gravity. 


SOLA  R   SYSTEM. 


CHAPTER    XI. 

A    GENERAL    DESCRIPTION    OF    THE    PLANETS. 

(  133.)  WE  conclude  this  section  of  astronomy  by  a  brief  CHAP.  XI. 
description  of  the  solar  system,  which  we  have  purposely 
delayed  lest  we  might  interrupt  the  course  of  reasoning 
respecting  the  planetary  motions.  The  reader  is  referred  to 
Table  III,  for  a  concise  and  comparative  view  of  all  the  facts 
that  can  be  numerically  expressed ;  and  aside  from  these  facts, 
little  can  be  said  by  way  of  explanation  or  description. 

The  fact,  that   the  sun  or  a  planet  revolves  on  an  axis,  Facts  reveal- 
must  be  determined  by  observing  the  motion  of  spots  on  the  onthe«nn0tl 
visible  disc  ;  and  if  no  spots, -are  visible,  the  fact  of  revolution  pianeu. 
cannot  be  ascertained.*     But  when   spots  are  visible,  their 
motion  and  apparent  paths  will  not  only  point  out  the  time 
of  revolution,  but  the  position  of  the  axis. 

THE     SUN. 

( 134.)  The  sun  is  the  central  body  in  the  system,  of  im-    The  sun  the 
mense  magnitude,  comparatively  stationary,  the  dispenser  of  reP°sitory  °f 
light  and  heat,  and  apparently  the  repository  of  that  force 
which  governs  the  motion  of  all  other  bodies  in  the  system. 

"Spots  on  the  sun  seem  first  to  have  been  observed  in  the  year  1611, 
since  which  time  they  have  constantly  attracted  attention,  and  have 
been  the  subject  of  investigation  among  astronomers.  These  spots 
change  their  appearance  as  the  sun  revolves  on  its  axis,  and  become 
greater  or  less,  to  an  observer  on  the  earth,  as  they  are  turned  to,  or 
from  him ;  they  also  change  in  respect  to  real  magnitude  and  number, 
one  spot,  seen  by  Dr.  Herschel,  was  estimated  to  be  more  than  six 
times  the  size  of  our  earth,  being  50000  miles  in  diameter.  Some- 
times forty  or  fifty  spots  may  be  seen  at  the  same  time,  and  sometimes 
only  one.  They  are  often  so  largo  as  to  be  seen  with  the  naked  eye ; 
this  was  the  case  in  1816. 

"  In  two  instances,  these  spots  have  been  seen  to  burst  into  several 
parts,  and  the  parts  to  fly  in  several  directions,  like  a  piece  of  ice 
thrown  upon  the  ground. 

*  Mercury  is  an  exception  to  this  principle. 


112  ASTRONOMY. 

CHAI-.  XI.  "  In  respect  to  the  nature  and  design  of  these  spots,  almost  every 
astronomer  has  formed  a  different  theory.  Some  have  supposed  them 
to  be  solid  opaque  masses  of  scoritu,  floating  in  the  liquid  fire  of  the 
sun  ;  others  as  satellites,  revolving  round  him,  and  hiding  his  light 
from  us;  others  as  immense  masses,  which  have  fallen  on  his  disc,  and 
which  are  dark  colored,  because  they  have  not  yet  become  sufficiently 
heated. 

"  Dr.  Herschel,  from  many  observations  with  his  great  telescope, 
concludes,  that  the  shining  matter  of  the  sun  consists  of  a  mass  of 
phosphoric  clouds,  and  that  the  spots  on  his  surface  are  owing  to  dis- 
turbances in  the  equilibrium  of  this  luminous  matter,  by  which  open- 
ings are  made  through  it.  There  are,  however,  objections  to  this 
theory,  as  indeed  there  are  to  all  the  others,  and  at  present  it  can  only  ba 
said,  that  no  satisfactory  explanation  of  the  cause  of  these  spots  has 
been  given." 

Singular      (  135.)    Mercury.     This  planet  is  the  nearest  to  the  sun, 
••••••rdlfr  an(j  jiag  keen  ^Q  gukject  Of  considerable  remark  in  the  pre- 

covenng    ro-  m  ....  . 

ceding  pages      It  is  rarely  visible,  owing  to  its  small  size  and 


proximity  to  the  sun,  and  it  never  appears  larger  to  the  na- 
ked eye  than  a  star  of  the  fifth  magnitude. 

Mercury  is  too  near  the  sun  to  admit  of  any  observations 
on  the  spots  on  its  surface  ;  but  its  period  of  rotation  has 
been  determined  by  the  variations  in  its  horns  —  the  same 
ragged  corner  comes  round  at  regular  intervals  of  time  — 
24h.  5m. 

Times  when  The  best  time  to  see  Mercury,  in  the  evening,  is  in  the 
Mercury  may  spring  of  the  year,  when  the  planet  is  at  its  greatest  elonga- 
tion east  of  the  sun.  It  will  then  be  visible  to  the  naked 
eye  about  fifteen  minutes,  and  will  set  about  an  hour  and 
fifty  minutes  after  the  sun.  When  the  planet  is  west  of  the 
sun,  and  at  its  greatest  distance,  it  may  be  seen  in  the  morn- 
ing, most  advantageously  in  August  and  September.  The 
symbol  for  the  greatest  elongation  of  Mercury,  as  written  in 
the  common  almanacs,  is  $  Gr.  Elon. 

High  moan  (  136.)  Venus.  This  planet  is  second  in  order  from  the  sun, 
tains  on  v«  an(j  jn  relation  to  its  position  and  motion,  has  been  sufficiently 
described.  The  period  of  its  rotation  on  its  axis  is  23h.  21m. 
The  position  of  the  axis  is  always  the  same,  and  is  not  at 
right  angles  to  the  plane  of  its  orbit,  which  gives  it  a  change  of 
seasons.  The  tangent  position  of  the  sun's  light  across  thif 


SOLAR   SYSTEM.  143 

planet  shows  a  very  rough  sur-  ffl|HH8^BHBHBB^B  CH4P-  **• 
face;  indeed,  high  mountains.  W3H,  i^r^m  :  •••-'"  T«i«copw 
By  the  radiating  and  glimmer-  »«•  •  »iew«ofv«- 

ing  nature  of  the  light  of  this 
planet,  we  infer  that  it  must 
have  a  deep  and  dense  atmos- 
phere. 

( 137.)  TJie  Earth  is  the  next  planet  in  the  system;  but  it    Th«  earth 
would  be  only  formality  to  give  any  description  of  it  in  this  a  planet 
place.      As  a'  planet,  it  seems  to  be  highly  favored  above  its 
neighboring  planets,  by  being  furnished  with  an  attendant,   The  earth'* 
the  moon;  and  insignificant  as  this  latter  body  is,  compared  at 
to  the  whole  solar  system,  it  is  the  most  important  and  in- 
teresting to  the  inhabitants  of  our  earth.      The  two  bodies, 
the  earth  and  the  moon,  as  seen  from  the  sun,  are  very  small : 
the  former  subtending  an'  angle  of  about  17"  in  diameter, 
the  latter  about  4",  and  their  distance  asunder  never  greater 
than  between  seven  and  eight  minutes  of  a  degree. 

Contrary  to  the  general  impression,  the  moon's  motion  in 
absolute  space  is  always  concave  toward  the  sun.* 

( 138.)  Mars  — the  first  superior  planet — is  of  a  red  color,    M*rt ;  «• 

......  .,     ,  .  ,  physical    an. 

and  very  variable  m  its  apparent  magnitude.     About  every  pearancefto 


*  This  may  be  shown  thus  —  the  moon  is  inside  the  earth's 
orbit  from  the  last  quarter  to  the  first  quarter,  on  an  average 
14  days  and  18  hours.  During  this  time  the  earth  moves  in 
its  orbit  14°  30'.  Let 
L  n  F  be  a  portion  of  the 
earth's  orbit  equal  to  14°  30', 
L  the  position  of  the  earth  at  the  First  Quarter  of  the  moon, 
and  F  its  position  at  the  Last  Quarter.  Draw  the  chord  L  F, 
and  compute  mn  the  versed  sine  of  the  arc  7°  15'. 

The  mean  radius  of  the  earth's  orbit  is  397  times  the  ra- 
dius of  the  lunar  orbit.  A  radius  of  397  and  an  angle  7°  15' 
givqs  a  versed  sine  of  3.17;  but  on  this  scale  the  distance 
from  the  earth  to  the  moon  is  unity,  or  less  than  one  third  of 
nm:  hence,  the  moon's  path  must  lie  between  the  chord  LF 
and  the  arc  L  n  F — that  is,  always  concave  toward  ike  turn. 


144  ASTRONOMY. 

CHAP^XI  other  year,  when  it  comes  to  ;the  meridian  near  midnight,  ifc 
is  then  most  conspicuous ;  and  the  next  year  it  is  scarcely 
noticed  by  the  common  observer. 

.    Tr.  "The    physical    appearance   of 
Telescopic  View  of  Mars.           ,,       .  , 
. Mars  is  somewhat  remarkable    His 

polar  regions,  when  seen  through 
a  telescope,  have  a  brilliancy  so 
much  greater  than  the  rest  of  his 
disc,  that  there  can  be  little  doubt 
that,  as  with  the  earth  so  with 
this  planet,  accumulations  of  ice 
or  snow  take  place  during  the  wm- 
ters  of  those  regions.  In  1781 
the  south  polar  spot  was  extremely 
bright ;  for  a  year  it  had  not  been 
exposed  to  the  solar  rays,  The 
color  of  the  planet  most  probably 
arises  from  adense  atmosphere  which  surrounds  him,  of  the  existence  of 
which  there  is  other  proof  depending  on  the  appearance  of  stars  as 
they  approach  him  ;  they  grow  dim  and  are  sometimes  wholly  extin- 
guished as  their  rays  pass  through  that  medium." 

Apparent im.      C  139 •)  The  next  planet,  as  known  to  ancient  astronomers* 

perfection  in  is  Jupiter ;  but  its  distance  is  so  great  beyond  the  orbit  of 

the  system.    jyfars^  fa^  ^  voj(j  Space  between  the  two  had  often  been 

considered  as  an  imperfection,  and  it  was  a  general  impression 

among  astronomers  that  a  planet  ought  to  occupy  that  vacant 

space. 

Bode'siaw.  Professor  Bode,  of  Berlin,  on  comparing  the  relative  dis- 
tances of  the  planets  from  the  sun,  discovered  the  following  re- 
markable fact — that  if  we  take  the  following  series  of  numbers : 

0,  3,   6,  12,  24,  48,     96,  192,  &c , 

and  then  add  the  number  4  to  each,  and  we  have, 

4,  7,  10,  16,  28,  52,  100,  196,  &o.f 

rhe  reason  and  this  last  series  of  numbers  very  nearly,  though  not  ex- 
•Mb!  cabled  ac%>  corresponds  to  the  relative  distances  of  the  planets  from 
•  (aw.          the  sun,  with   the  exception  of  the  number  28.      This  is 
sometimes  called  Bode's  law ;  but  remarkable  as  it  certainly 
is,  it  should  not  be  dignified  by  the  term  law,  until  some  bet- 
ter account  of  it  can  be  given  than  its  mere  existence ;  for, 
at  present,  all  that  can  be  said  of  it  is,  "  here  is  an  astonishing 


SOLAR  SYSTEM. 


145 


coincidence."     But,  mere  accident  as  it  may  be,  it  suggested    CHAP.  XL 
the  possibility  of  some  small,  undiscovered  planet  revolving    A~bo!d~hj. 
in  this  region,  and  we  can  easily  imagine  the  astonishment  of  pothesu 
astronomers,   on   finding  four  in   place  of  one,  revolving  in 
orbits  tolerably  well  corresponding  to  this  law,  or  rather  co- 
incidence.    Had  they  even  found  but  one,  it  would  seem  to 
indicate  something  more  than  nere  coincidence ;  but  finding 
four,  proves  the  series  to  be  simply  accidental  —  unless  the 
four  or   more  planets   there    discovered  were  originally  one 
planet ;  and  then  came  the  inquiry,  is  not  this  the  case  ?  Thus 
originated  the  idea  that  these  new  and  newly  discovered  small 
planets  are  but  fragments  of  a  larger  one,  which  formerly  cir- 
culated in  that  interval,  and  was  blown  to  pieces  by  some 
internal  explosion  —  and  we  shall  examine  this  hypothesis  in  a 
text  note,  under  physical  astronomy. 

The  names  of  these  planets,  in  the  order  of  the  times  of  their 
discovery,  are,  Ceres,  Pallas,  Juno,  Vesta.  The  order  of  their 
distances  from  the  sun,  is  Vesta,  Juno,  Ceres,  Pallas 


Planets. 

Names  of  Dis- 
coverers. 

Residence  of  Discoverers. 

Date  of  Discovery. 

1  Ceres... 
Pallas... 
j  Juno  .  .  . 
|  Vesta... 

M.  Piazzi, 
Dr.  Olbers, 
M.  Harding', 
Dr.  Olbers, 

Palermo,  Sicily, 
Bremen,  Germany, 
Lilienthal,  near  Bremen, 
Bremen, 

1st  Jan.,  1801. 
28th  Mar.,  1802. 
1st  Sept.  1804. 
29th  Mar.,  1807. 

If  a  planet  has  really  burst,  it  is  but  reasonable  to  suppose 
that  it  separated  into  many  fragments ;  and,  agreeably  to  this 
view  of  the  subject,  astronomers  have  been  constantly  on  the 
alert  for  new  planets,  in  the  same  regions  of  space ;  and  every         Recent 
discovery  of  the  kind  greatly  increases  the  probability  of  the  discoveries 
theory.     The  following  very  recent  discoveries  are  said  to  have 
been  made,  but  the  elements  of  the  orbits  are  not  regarded  as 
sufficiently  accurate  to  demand  a  place  in  the  table. 

On  the  8th  of  December,  1845,  Mr.  Hencke,  of  Dreisen, 
claims  to  have  discovered  a  planet  which  he  calls  Astrea; 
and  the  same   observer   also    claims  another,  discovered  in    New  plan 
1847,  called  Hele.     His  success  induced  others  to  a  like  exa-  et«  <«»«>vef 
mination,  and  a  Mr.  Hind,  of  London,  within  the  past  year, 
10 


146 


ASTRONOMY. 


CHAP,  xi  1848,  claims  a  seventh  and  eighth  asteroid,  iiai.aed  Iris  and 
Flora. 

Thus  we  have  eight  miniature  worlds,  supposed  to  have 
once  composed  a  planet ;  and  if  the  four  last  named  are  veri- 
table discoveries,  we  shall  soon  have  the  elements  of  their 
orbits  in  an  unquestionable  shape. 

The  elements  of  the  orbits  of  the  four  known  asteroids,  la 
given  for  the  epoch  1820,  are  not  as  accurate  as  the  follow- 
ing, which  were  deduced  from  the  Nautical  Almanac  for  1846 
and  1847 ;  which  have  been  corrected  from  more  modern, 
extended,  and  accurate  observations.  (Epoch  Jan.,  1847.) 

On  account  of  the  small  magnitude  of  these  new  planets, 
and  their  recent  discovery,  nothing  is  known  of  them  save 
the  following  tabular  facts,  and  these  are  only  approximation 
to  the  truth. 


Planets. 

Sidereal 
Revolutions. 

Mean  Distance  from 
the  Sun. 

Eccentricity  of 
Orbits. 

Vesta  

Days. 
1324.  289 

2  36120 

0.  08913 

1594.  721 

2  66514 

0.  25385 

Ceres  

1683  064 

2  76910 

0  07844 

Pallas    . 

1685  162 

2  77125 

0  24050 

Planets. 

Longitude  of 
Ascending  Node. 

Inclination  of 
Orbits. 

Longitude  of 
Perihelion. 

Vesta  

O        '        " 

103  ^o  47 

O      '        " 

7     8  29 

0       »        " 

251     4  34 

Juno    ...... 

170  53     0 

13     2  53 

54   18  32 

Ceres  

80  47   56 

10  37   17 

147  25  41 

Pallas  

172  42  38 

34  37  42 

121   20   13 

object  of  ( 140.)  With  the  two  elements,  the  longitude  of  the  ascend- 
Fif .  29.  ing  nodes,  and  the  inclination  of  the  orbits  to  the  ecliptic,  we 
are  enabled  to  give  a  general  projection  of  these  orbits  around 
the  celestial  sphere,  in  relation  to  the  ecliptic,  as  represented 
on  page  37 ;  and  our  object  is  to  show  that  there  are  two 
points  in  the  heavens,  nearly  opposite  to  each  other,  near  to 
which  all  these  planets  pass.  One  of  these  points  is  about 
the  longitude  of  185  degrees,  and  the  latitude  of  15  degrees 
north ;  and  the  other  is  the  opposite  point  on  the  celestial 
sphere.  If  these  planets  are  but  fragments  of  one  original 
planet,  which  burst  or  exploded  by  its  internal  fires,  from  that 


SOLAR    SYSTEM 


moment  they  must  have 
started  from  the  same 
point,  andi/<e  orbits  of  all 
have  one  common  distance 
from  the  sun ;  and  for 
ages  after  such  a  catas- 
trophe, these  fragments 
must  have  had  nearly  a 
common  node  j  and  the 
fact  that  they  do  not,  at 
present,  pass  through  a 
common  point,  nor  have 
a  common  node,  does  not 
prove  that  they  were  not 
originally  in  one  body; 
tor,  owing  to  mutual  dis- 
turbances, and  the  dis- 
turbances of  other  pla- 
nets, the  nodes  must 
change  positions;  and  the 
longer  axis  of  the  orbits, 
especially  the  very  ec- 
centric ones,  must  change 
positions ;  and  now  (after 
we  know  not  how  many 
ages),  it  is  not  incon- 
sistent with  the  theory 
of  an  explosion,  that  we 
mid  the  orbits  as  they 
are. 

The  hypothesis  that 
these  planets  were  ori- 
ginally one,  and  must, 
therefore,  have  two  com- 
mon points  in  the  hea- 
vens near  which  they 
must  all  pass,  led  to  the 
iixcovery  of  Juno  and 


147 

CHAP.  Xi. 

Where  th« 

original  pla- 
net must 
have  explod- 
ed,  if  the 
hypothesis 
of  an  original 
plamn  is  mi» 


148  ASTRONOMY. 

CHAP.  M.   Vesta,    by   carefully   observing  these   two   portions  of   the 
heavens. 

The  apparent  diameters  of  these  planets  are  too  small  to 
be  accurately  measured;  and  therefore  we  have  only  a  very 
rough  or  conjectural  knowledge  of  their  real  diameters. 

All  of  these  planets  are  invisible  to  the  naked  eye,  except 
Vesta,  which  sometimes  can  be  seen  as  a  star  of  the  5th  or 
6th  magnitude. 

(141.)  Jupiter.  We  now  come  to  the  most  magnificent 
planet  in  the  system  —  the  well-known  Jupiter — which  is 
nearly  1300  times  the  magnitude  of  the  earth. 
Japiter'«  The  disc  of  Jupiter  is  always  observed  to  be  crossed,  in  an 
eastern  and  western  direction,  by  dark  bands,  as  represented 
in  Fig.  30. 

Fig.  30.  —  Telescopic  View  of  Jupiter. 


41  These  belts  are,  however,  by  no  means  alike  at  all  times ;  they 
vary  in  breadth  and  in  situation  on  the  disc  (though  never  in  their 
general  direction).  They  have  even  been  seen  broken  up,  and  distri- 
buted over  the  whole  face  of  the  planet :  but  this  phenomenon  is  ex- 
tremely rare.  Branches  running  out  from  them,  and  subdivisions,  as 
represented  in  the  figure,  as  well  as  evident  dark  spots,  like  strings  of 
clouds,,  are  by  no  means  uncommon  ;  and  from  these,  attentively 
watched,  it  is  concluded  that  this  planet  revolves  in  the  surprisingly 
Diurnal  re-  short  period  Of  9  n.  55m.  50s.  (sid.  time),  on  an  axis  perpendicular  to 
the  direction  of  the  belts.  Now,  it  is  very  remarkable,  and  forms  a 
most  satisfactory  comment  on  the  reasoning  by  which  the  spheroidal 
figure  of  the  earth  has  been  deduced  from  its  diurnal  rotation,  that  the 
outline  of  Jupiter's  disc  is  evidently  not  circular,  but  elliptic,  being 
considerably  flattened  in  the  direction  of  its  axis  of  rotation. 


SOLAR    SYSTEM. 


i49 


*'  The  parallelism  of  the  belts  to  the  equator  of  Jupiter,  their  occa-   CHAP.  XL 

sional  variations,  and  the  appearances  of  spots  seen  upon  them,  render     

it  extremely  probable  that  they  subsist  in  the  atmosphere  of  the  planet,  *  » 
forming  tracts  of  comparatively  clear  sky,  determined  by  currents  ana- 
logous to  our  tradewinds,  but  of  a  much  more  steady  and  decided  cha- 
racter, as  might  indeed  be  expected  from  the  immense  velocity  of  its 
rotation.  That  it  is  the  comparatively  darker  body  of  the  planet  which 
appears  in  the  belts,  is  evident  from  this, —  that  they  do  not  come  up 
iu  all  their  strength  to  the  edge  of  the  disc,  but  fade  away  gradually  be- 
fore they  reach  it. 

(142.)  "When  Jupiter  is  viewed  with  a  telescope,  even  of  moderate  Jupiter'i 
power,  it  is  seen  accompanied  by  four  small  stars,  nearly  in  a  straight  *ateHites 
line  parallel  to  the  ecliptic.  These  always  accompany  the  planet,  and 
are  called  its  Satellites.  They  are  continually  changing  their  positions 
with  respect  to  one  another,  and  to  the  planet,  being  sometimes  all  to 
the  right,  and  sometimes  all  to  the  left ;  but  more  frequently  some  on 
each  side.  The  greatest  distances  to  which  they  recede  from  the  planet, 
on  each  side,  are  different(  for  the  different  satellites,  and  they  are  thus 
distinguished  :  that  being  called  the  First  satellite,  which  recedes  to  the 
least  distance  ;  that  the  Second,  which  recedes  to  the  next  greater  dis- 
tance, and  so  on.  The  satellites  of  Jupiter  were  discovered  by  Galileo, 
in  1610. 

"  Sometimes  a  satellite  is  observed  to  pass  between  the  sun  and  Ju- 
piter, and  to  cast  a  shadow  which  describes  a  chord  across  the  disc. 
This  produces  an  eclipse  of  the  sun,  to  Jupiter,  analogous  to  those 
which  the  moon  produces  on  the  earth.  It  follows  that  Jupiter  and 
its  satellites  are  opake  bodies,  which  shine  by  reflecting  the  sun's 
light. 

*'  Careful  and  repeated  observations  show  that  the  motions  of  the  satel- 
lites are  from  west  to  east,  in  orbits  nearly  circular,  and  making  small 
angles  with  the  plane  of  Jupiter's  orbit.  Observations  on  the  eclipses 
of  the  satellites  make  known  their  synodic  revolutions,  from  which 
their  sidereal  revolutions  are  easily  deduced.  From  measurements  of 
the  greatest  apparent  distances  of  the  satellites  from  the  planet,  their 
real  distances  are  determined. 

"  A  comparison  of  the  mean  distances  of  the  satellites,  with  their  side- 
leal  revolutions,  proves  that  Kepler's  third  law,  with  respect  to  the 
planets,  applies  also  to  the  satellites  of  Jupiter.  The  squares  of  their 
sidereal  revolutions  are  as  the  cubes  of  their  mean  distances  from  the 
planet. 

"  The  planets  Saturn  and  Uranus  are  also  attended  by  satellites,  and 
the  same  law  has  place  with  them." 

(  143.)  By  the  eclipses  of  Jupiter's  satellites,  the  progres- 
give  nature  of  light  was  discovered ;  which  we  illustrate  in  light, 
the  following  manner : 


ASTRONOMY 
Fig.  31. 


Lot  S  (Fig.  31)  represent  the  sun,  J  Jupiter,  £  earth,  and  m  Jupiter'i 
first  satellite.  By  careful  and  accurate  observations  astronomers  havo 
decide,!  that  the  mean  revolution  of  this  satellite  round  its  primary,  is 
performed  in  42  li.  23  m  and  35s.;  that  is,  the  mean  time  from  one 
eclipse  to  another. 

Velocity  oi       But  when  the  earth  is  at  E,  and  moving  in  a  direction  toward,  or 
light,      how  nearly  toward,  the  planet  as  represented  in  the  figure,  the  mean  time 
iatemuned.    between  two  consecutive  eclipses  is  shortened  about  15  seconds;  and 
we  can  explain  this  on  no  other  hypothesis  than  that  the  earth  has  ad- 
vanced and  met  the  successive  progression  of  light.     When  the  earth 
is  in  position  as  respects  the  sun  and  Jupiter,  as  represented  in  our 
figure  at  E',  ana  moving  from  Jupiter,  then  the  interval  between  two 
consecutive  eclipses  of  Jupiter's  first  satellite  is  prolonged  or  increased 
about  15  seconds. 

But  during  the  interval  of  one  revolution  of  Jupiter's  first  satellite, 
the  earth  moves  in  its  orbit  about  2880000  miles  ;  this,  divided  by  15, 
gives  I920f)0  miles  for  tiie  motion  of  light  in  one  second  of  time  ;  and 
this  velocity  will  carry  light  from  the  sun  to  the  earth  in  about  eight 
and  one-fourth  minutes. 

Longitude      ( 144. )  As  an  eclipse  of  one  of  Jupiter's    satellites  maybe 

found  by  the  seen  from  a]j  p]aces  where  the  planet  is  there  visible,  two 

Jupiter's  »°a-  observers  viewing  it  will  have  a  signal  for  the  same  moment, 

toiiites.        at  their  respective  places ;  and  their  difference  in  local  time 

will  give  their  difference  in  longitude.     For  example,  if  one 

observer  saw  one  of  these  eclipses  at  10  h.in  the  evening,  and 

another  at  8  h.  30  m.,  the  difference  of  longitude  between  the 

observers  would  be  1  h.  30  m.  in  time,  or  22°  30'  of  arc. 

The  absolute  time  that  the  eclipse  takes  place,  is  the  same 
to  all  observers ;  and  he  who  has  the  latest  local  time  is  the 
most  eastward. 

These  eclipses  cannot  be  observed  at  sea,  by  reason  of  the 
motion  of  the  vessel. 


SOLAR  SYSTEM.  15] 

(145.)  Saturn.     The  next  planet  in  order  of  remoteness  CHAP.  XL 
from  the,  sun,  is  Saturn,  the  most  wonderful  object  in  the      Satum- 
solar  system.     Though  less  than  Jupiter,  it  is  about  79000  *****•• 
miles  in  diameter,  and  1000  times  greater  than  our  earth. 

"  This  stupendous  globe,  besides  being  attended  by  no  less  than  seven 
satellites,  or  moons,  is  surrounded  with  two  broad,  flat,  extremely  thin 
rings,  concentric  with  the  planet  and  with  each  other  ;  both  lying  in 
one  plane,  and  separated  by  a  very  narrow  interval  from  each  other 
throughout  their  whole  circumference,  as  they  are  from  the  planet  by 
a  much  wider.  The  dimensions  of  this  extraordinary  appendage  are 
as  follows  : 

Exterior  diameter  of  exterior  ring, =  176418. 

Interior  ditto, =  155272. 

Exterior  diameter  of  interior  ring, =  151690. 

Interior  ditto, =  117339. 

Equatorial  diameter  of  the  body, =     79160 

Interval  between  the  planet  and  interior  ring, =     19090. 

Interval  of  the  rings    =       1791. 

Thickness  of  the  rings  not  exceeding, =         100.        Dimension! 

Fig.  32.  —  Telescopic  View  of  Saturn.  °f  **  rmgS* 


"  The  figure  represents  Saturn  surrounded  by  its  rings,  and  having  its  The  ringi 
body  striped  with  dark  belts,  somewhat  similar,  but  broader  and  less  ar* 
strongly  marked  than  those  of  Jupiter,  and  owing, doubtless,  to  a  simi- 
!ar  cause.  That  the  ring  is  a  solid  opake  substance,  is  shown  by  its 
throwing  its  shadow  on  the  body  of  the  planet,  on  the  side  nearest  the 
sun,  and  on  the  other  side  receiving  that  of  the  body,  as  shown  in  the 
figure.  From  the  parallelism  of  the  belts  with  the  plane  of  the  ring, 
it  may  be  conjectured  that  the  axis  of  rotation  of  the  planet  is  perpen- 
dicular to  that  plane  ;  and  this  conjecture  is  confirmed  by  the  occa- 
sional appearance  of  extensive  dusky  spots  on  its  surface,  which  when 
vatched,  like  the  spots  on  Mars  or  Jupiter,  indicate  a  rotation  in  10  h. 
29  in.  17  s.  about  an  axis  so  situated. 

"  It  will  naturally  be  asked  how  so  stupendous  an  arch,  if  composed 
of  solid  and  ponderous  materials,  can  be  sustained  without  collapsing 


.  »i  ASTRONOMY. 

CHAP.  XI.    and  falling  in  upon  the  planet  ?     The  answer  to  this  is  to  be  found  in 

The  stabi-  a  sw''1  "°lat>o11  °f  the  r'ng  'n  lia  own  plane,  which  observation  haa 

lity    of    the  detected,  owing  to  some  portions  of  the  ring  being  a  little  less  bright 

riafcs.  than  others,  and  assigned  its  period  at  10  h.  29m.  17  8.,  which,  from 

what  we  know  cf  its  dimensions,  and  of  the  force  of  gravity  in  the 

Saturnian  system,  is  very  nearly  the  periodic  time  of  a  satellite  revolv- 

ing at  the  same  distance  as  the  middle  of  its  breadth.     It  is  the  centri- 

fugal force,  then,  arising  from  this  rotation,  which  sustains  it  ;  and, 

although  no  observation  nice  enough  to  exhibit  a  difference  of  periods 

between  the  outer  and  inner  rings  have  hitherto  been  made,  it  is  more 

than  probable  that  such  a  difference  does  subsist  as  to  place  each  inde- 

pendently of  the  other  in  a  similar  state  of  equilibrium. 

The  rings  "  Although  the  rings  are,  as  we  have  said,  very  nearly  concentric 
revolve  a-  wjth  the  body  of  Saturn,  yet  recent  micrometrical  measurements,  of 
extreme  delicacy,  have  demonstrated  that  the  coincidence  is  not  inathe- 
*  mat'ca"y  exact,  but  that  the  center  of  gravity  of  the  rings  oscillates 
round  that  of  the  body,  describing  a  very  minute  orbil,  probably  under 
laws  of  much  complexity.  Trifling  as  this  remark  may  appear,  it  is 
of  the  utmost  importance  to  the  stability  of  the  system  of  the  rings. 
Supposing  them  mathematically  perfect  in  their  circular  form,  and 
exactly  concentric  with  the  planet,  it  is  demonstrable  that  they  would 
form  (  in  spite  of  their  centrifugal  force  )  a  system  in  a  state  of  unstable 
equilibrium,  which  the  slightest  external  power  would  subvert  —  not  by 
causing  a  rupture  in  the  substance  of  the  rings  —  but  by  precipitating 
them,  unbroken,  on  the  surface  of  the  planet.  For  the  attraction  of 
such  a  ring  or  rings  on  a  point  or  sphere  eccentrically  situate  within 
them,  is  not  the  same  in  all  directions,  but  tends  to  draw  the  point  or 
sphere  toward  the  nearest  part  of  the  ring,  or  away  from  the  center. 
Hence,  supposing  the  body  to  become,  from  any  cause,  ever  so  little 
eccentric  to  the  ring,  the  tendency  of  their  mutual  gravity  is,  not  to 
correct,  but  to  increase  this  eccentricity,  and  to  bring  the  nearest  parts 
of  them  together." 

(146.)  Uranus.  The  next  planet,  beyond  Saturn,  was 
Henchei.  discovered  by  Sir  W.  F.  Herschel,  in  1781,  and,  for  a  time, 
was  called  Herschel,  in  honor  of  its  discoverer;  but,  accord- 
ing to  custom,  the  name  of  a  heathen  deity  has  been  substi- 
tuted, and  the  planet  is  now  called  Uranus  —  the  father  of 
Saturn. 

This  lanet      ^n*s  P^ane*i  'ls  rarely  to  be  seen,  without  a  telescope.     In  a 

rarely  visible  clear  night,  and  in  the  absence  of  the  moon,  when  in  a  favor- 

o  the  naked  ^^  pOSit,ion  above  the  horizon,  it  may  be  seen  as  a  star  of 

about  the  6th  magnitude.     Its  real  diameter  is  about  35000 

miles,  and  about  80  times  the  magnitude  of  the  earth. 


SOLAR   SYSTEM.  153 

The  existence  of  this  planet  was  suggested  by  some  CHAP.  XL 
of  the  perturbations  of  Saturn  ;  which  could  not  be  accounted 
for  by  the  action  of  the  then  known  planets  ;  but  it  does  not 
appear  that  any  computations  were  made,  as  a  guide  to  the 
place  where  the  unknown  disturbing  body  ought  to  exist  ;  and, 
as  far  as  we  know,  the  discovery  by  Herschel  was  ?nere 
accident. 

But  not  so  with  the  planet  Neptune,   discovered   in  the    Facts   led 


latter  part  of  September,  1846,  by  a  French  astronomer,  Le-  to  the  'Hs 
terrier  ;  and  also  a  Mr.  Adams,  of  Cambridge,  England,  who  has  tune 
put  in  his  claim  as  the  discoverer.  The  truth  is,  that  the 
attention  of  the  astronomers  of  Europe  had  been  called  to 
some  extraordinary  perturbations  of  Uranus  ;  which  could  not 
be  accounted  for  without  supposing  an  attracting  body  to  be 
situated  in  space,  beyond  the  orbit  of  Uranus  ;  and  so  distinct 
and  clear  were  these  irregularities,  that  both  geometers,  Le- 
verrier  and  Adams,  fixed  on  the  same  region  of  the  heavens, 
for  the  then  position  of  their  hypothetical  planet  ;  and  by  dili- 
gent search,  the  planet  was  actually  discovered  about  the 
game  time,  in  both  France  and  England. 

At  present,  we  can  know  very  little  of  this  planet  ;  and 
according  to  the  best  authority  I  can  gather,  its  longi- 
tude, January  1,  1847,  was  327°  24'.  Mean  distance  from 
the  sun,  30.2  (  the  earth's  distance  being  unity)  ;  period  of 
revolution  166  years.  Eccentricity  of  orbit  0.0084;  mass, 

1 
23000"' 

According  to  Bode's  law,  the  distance  of  the  next  planet 
from  the  sun,  beyond  Uranus,  must  be  38.8  ;  and  if  Neptune 
really  is  at  30.2,  it  shows  Bode's  law  to  be  only  a  remarkable 
coincidence  ;  for  there  can  be  no  exceptions  to  positive  physi- 
cal laws. 

"  We  shall  close  this  chapter  with  an  illustration  calculated  to  convey  How  u 
to  the  minds  of  our  readers  a  general  impression  of  the  relative  magni-  Obtain  a  t  »r- 
tudes  and  distances  of  the  parts  of  our  system.  Choose  any  well-  rect  concep- 
leveled  field  or  bowling  green.  On  it  place  a  globe,  two  feet  in  diame-  tion  of  the 
ter  ;  this  will  represent  the  sun  ;  Mercury  will  be  represented  by  a  grain  solar  system 
of  mustard  seed,  on  the  circumference  of  a  circle  164  feet  in  diameter, 
for  its  orbit;  Venus  a  pea,  on  a  circle  284  feet  in  diameter  ;  the  earth 


154  ASTRONOMY. 

CHAP.  XI.  also  a  pea,  on  a  circle  of  430  feet;  Mars  a  rather  large  pin's  head,  on  a 
circle  of  654  feet;  Juno,  Ceres,  Vesta,  and  Pallas,  grains  of  sand,  in 
orbits  of  from  1000  to  1200  feet ;  Jupiter  a  moderate-sized  orange,  in  a 
circle  nearly  half  a  mile  across;  Saturn  a  small  orange,  on  a  circle  of 
four-fifths  of  a  mile  ;  and  Uranus  a  full-sized  cherry,  or  small  plum, 
upon  the  circumference  of  a  circle  more  than  a  mile  and  a  half  in  dia- 
meter. As  to  getting  correct  notions  on  this  subject  by  drawing  circles 
on  paper,  or,  still  worse,  from  those  very  childish  toys  called  orreries, 
it  is  out  of  the  question.  To  imitate  the  motions  of  the  planets  in  the 
View  of  above-mentioned  orbits,  Mercury  must  describe  its  own  diameter  in  41 

the  planetary  seconds;  Venus,  in  4  m.  14s. ;  the  earth,  in  7  minutes  ;  Mars,  in  4m. 

motions.  48S.  .  Jupiter,  in2h  56m. ;  Saturn,  in  3  h.  13m. ;  and  Uranus,  in  2h. 
16  m." — HerscheVs  Astronomy. 


CHAPTER   XII. 

ON     COMETS. 

CHAP.  xn.       (147.)  BESIDES  the  planets,  and  their  satellites,  there  are 
Comets  great  numbers  of  other  bodies,  which  gradually  come  into 

formerly   in-  view,  increasing  in  brightness  and  velocity,  until  they  attain 

ror          "    a  maximum,  and  then  as  gradually  diminish,  pass  off,  and  are 

lost  in  the  distance. 
Knowledge      "These  bodies  are  comets.     From  their  singular  and  unusual  appear- 

b-irtishes        ance,  they  were  for  a  long  time  objects  of  terror  to  mankind,  and  were 

dread.  regarded  as  harbingers  of  some  great  calamity. 

"  The  luminous  train  which  accompanied  them  was  particularly 
alarming,  and  the  more  so  in  proportion  to  its  length.  It  is  but  little 
more  than  half  a  century  since  these  superstitious  fears  were  dissipated 
by  a  sound  philosophy  ;  and  comets,  being  now  better  understood, 
excite  only  the  curiosity  of  astronomers  and  of  mankind  in  general. 
These  discoveries  which  give  fortitude  to  the  human  mind  are  not 
among  the  least  useful. 

"  It  was  formerly  doubted  whether  comets  belonged  to  the  class  of 
heavenly  bodies,  or  were  only  meteors  engendered  fortuitously  in  the 
air  by  the  inflammation  of  certain  vapors.  Before  the  invention  of  the 
telescope,  there  were  no  means  of  observing  the  progressive  increase 
and  diminution  of  their  light.  They  were  seen  but  for  a  short  time, 
and  their  appearance  and  disappearance  took  place  suddenly.  Their 
light  and  vapory  tails,  through  which  the  stars  were  visible,  and  their 
whiteness  often  intense,  seemed  to  give  them  a  strong  resemblance  to 
those  transient  fires,  which  we  call  shooting  stars.  Apparently,  they 
differed  from  these  only  in  duration.  They  might  be  only  composed 


COMETS  155 

of  a  more  compact  substance  capable  of  retarding  for  a  longer  time   CHAP.  XIL 
their  dissolution.     But  these  opinions  are  no  longer  maintained  ;  more 
accurate  observations  have  led  to  a  different  theory. 

"All  the  comets  hitherto  observed  have  a  small  parallax,*  which  places    Parallax  of 
them  far  beyond  the  orbit  of  the  moon  ;  they  are  not,  therefore,  formed  comets, 
in  our  atmosphere.     Moreover,  their  apparent  motion  among  the  stars 
is  subject  to  regular  laws,  which  enable  us  to  predict  their  whole  course 
from  a  small  number  of  observations.     This  regularity  and  constancy 
evidently  indicate  durable  bodies  ;  and  it  is  natural  to  conclude  that 
corm-ts  are  as  permanent  as  the  planets,  but  subject  to  a  different  kind 
of  movement. 

"  W  hen  we  observe  these  bodies  with  a  telescope,  they  resemble  a  mass  Comets  are 
of  vapor,  at  the  center  of  which  is  commonly  seen  a  nucleus  more  or  aPParentl7 
less  distinctly  terminated.  Some,  however,  have  appeared  to  consist  m*r* 
of  merely  a  light  vapor,  without  a  sensible  nucleus,  since  the  stars  are 
visible  through  it.  During  their  revolution,  they  experience  progres- 
sive variatious  in  their  brightness,  which  appear  to  depend  upon  their 
distance  from  the  sun,  either  because  the  sun  inflames  them  by  its  heat, 
or  simply  on  account  of  a  stronger  illumination.  When  their  bright- 
ness is  greatest,  we  may  conclude  from  this  very  circumstance  that 
they  are  near  their  perihelion.  Their  light  is  at  first  very  feeble,  but 
becomes  gradually  more  vivid,  until  it  sometimes  surpasses  that  of  the 
brightest  planets  ;  after  which  it  declines  by  the  same  degrees  until  it 
becomes  imperceptible.  We  are  hence  led  to  the  conclusion  that 
comets,  coming  from  the  remote  regions  of  the  heavens,  approach,  ill 
many  instances,  much  nearer  the  sun  than  the  planets,  and  then  recede 
to  much  greater  distances. 

"Since  comets  are  bodies  which  seem  to  belong  to  our  planetary     Orbits    *f 
system,  it  is  natural  to  suppose  that  they  move  about  the  sun  like  comets 
planets,  but  in  orbits  extremely  elongated.     These  orbits  must,  there- 
fore, still  be  ellipses,  having  their  foci  at  the  center  of  the  sun,  but 
having  their  major  axes  almost  infinite,  especially  with  respect  to  us, 
who  observe  only  a  small  portion  of  the  orbit,  namely,  that  in  which 
the  comet  becomes  visible  as  it  approaches  the  sun.     Accordingly  the 
orbits  of  comets  must  take  the  form  of  a  parabola,  for  we  thus  designate 
the  curve  into  which  the  ellipse  passes,  when  indefinitely  elongated. 

"  If  we  introduce  this  modification  into  the  laws  of  Kepler,  which 

. 

*  The  parallaxes  of  comets  are  known  to  be  small,  by  two  observers, 
at  distant  stations  on  the  earth,  comparing  their  observations  taken 
or.  the  same  como .  at  near  the  same  time.  At  the  times  the  observa- 
tions are  made.,  neither  observer  can  know  how  great  the  parallax  is. 
It  is  only  afterward,  when  comparisons  are  made,  that  judgment,  in 
this  particular,  can  be  formed  ;  and  it  is  not  common  that  any  more 
definite  conclusion  can  be  drawn,  than  that  the  parallax  is  small,  and, 
of  course,  the  body  distant. 


156  ASTRONOMY. 


.  XII.  relate  to  the  elliptical  motion,  we  obtain  those  of  the  parabolic  motion 

of  comets. 

Comets  de§-  "  ^eace  '*  follows  that  the  areas  described  by  the  same  comet,  in  its 
cribe  equal  parabolic  orbit,  are  proportional  to  the  times.  The  areas  described  by 
areas  in  e-  different  comets  in  the  same  time,  are  proportional  to  the  square  root* 
qual  times.  of  their  perihelion  distances. 

"  Lastly,  if  we  suppose  a  planet  moving  in  a  circular  orbit,  whose 
radius  is  equal  to  the  perihelion  distance  of  a  comet,  the  areas  described 
by  these  two  bodies  in  the  same  time,  will  be  to  each  other  as  1  to 
/2.   Thus  are  the  motions  of  comets  and  planets  connected. 

"  By  means  of  these  laws  we  can  determine  the  area  described  bv 

a  comet  in  a  given  time  after  passing  the  perihelion,  and  fix  its  posi- 

tion in  the  parabola.     It  only  remains  then  to  bring  this  theory  to  th«i 

test  of  observation.     Now  we  have  a  rigorous  method  of  verifying  it, 

by  causing  a  parabola  to  pass  through  several   observed  places  of  a 

comet,  and  then  ascertaining  whether  all  the  others  are  contained  in  it 

Three  obser-      "  For  this  purpose  three  observations  are  requisite.      If  we  observe 

rations  suffi-  the   right  ascension   and  declination    of  a  comet  at   three   different 

cient  to  find  times,   and   thence  deduce  its  geocentric    longitude  and  latitude,  we 

the  orbit  of  a  8\m\\  nave  the  direction  of  three  visual  rays  drawn  at  these  times  from 

the  earth   to   the  comet,  and  in  the  prolongation  of  which  it  must 

necessarily  be  found.     The  corresponding  places  of  the  sun  are  also 

known  ;  it  remains  then  to  construct  a  parabola,  having  its  focus  at 

the  center  of  the  sun,  and  cutting  the  visual  rays  in  points,  the  inter- 

vals of  which  correspond  to  the  number  of  days  between  the  obser- 

vations. 

"  Or  if  we  suppose  the  earth  in  mo- 
tion and  the  sun  at  rest,  let  T,  T,  T", 
represent  three  successive  positions  of 
the  earth,  and  TC,  T'C,  T"C",  three 
visual  rays  drawn  to  the  comet.  The 
question  is  to  find  a  parabola  CC'C", 
having  its  focus  in  <S  at  the  center  of 
the  sun,  and  cutting  the  three  visual 
rays  conformably  to  the  conditions  re- 
quired. 

The  orbit  of  a      "  These  conditions  are  more  than  sufficient  to  determine  completely 

comet  found  the  elements  of  the  parabolic  motion,  that  is,  the  perihelion  distance 

by  three  ob-  of  the  comet,  the  position  of  the  perihelion,  the  instant  of  passing  this 

wrvations.     point,  the  inclination  of  the  orbit  to  the  ecliptic,  and  the  position  of 

its  nodes.     These  five  elements  being  known,  we  can  assign  the  posi- 

tion of  the  comet  for  any  time  whatever,  and  compare  it  with  the 

results  of  observation.     But  the  calculation  of  the  elements  is  very 

difficult,  and  can  be  performed  only  by  a  very  delicate  analysis,  which 

cannot  here  be  made  known. 


COMETS.  157 

"  About  120  comets  have  been  calculated  upon  the  theory  of  the   CHAP.  XII. 
parabolic  motion,  and  the  observed  places  are  found  to  answer  to  such 
a  supposition.     We  can  have  no  doubt,  therefore,  that  this  is  conform-    jnc]inatio 
able  to  the  law  of  nature.     We  have  thus  obtained  precise  knowledge  Of  tj,ejr  ^ 
of  the  motions  of  these  bodies,  and  are  enabled  to  follow  them  in  space,  bits. 
This  discovery  has  given  additional  confirmation  to  the  laws  of  Kepler, 
and  led  to  several  other  important  results. 

«'  Comets  do  not  all  move  from  west  to  east  like  the  planets.  Some 
have  a  direct,  and  some  a  retrograde  motion. 

"Their  orbits  are  not  comprehended  within  a  narrow  zone  of  the 
heavens,  like  those  of  the  principal  planets.  They  vary  through  all 
degrees  of  inclination.  There  are  some  whose  plane  is  nearly  coinci- 
dent with  that  of  the  ecliptic,  and  others  have  their  planes  perpendicular 
to  it. 

"  It  is  farther  to  be  observed  that  the  tails  of  comets  begin  to  appear, 
as  the  bodies  approach  near  the  sun  ;  their  length  increases  with  this 
proximity,  and  they  do  not  acquire  their  greatest  extent,  until  after 
passing  the  perihelion.  The  direction  is  generally  opposite  to  the  sun, 
forming  a  curve  slightly  concave,  the  sun  on  the  concave  side. 

"  The  portion  of  the  comet  nearest  to  the  sun  must  move  more  rapidly 
than  its  remoter  parts,  and  this  will  account  for  the  lengthening  of  the 
tail. 

"The   tail  is,  however,  by  no  means  an   invariable  appendage  of      Some  corn- 
comets.     Many  of  the  brightest  have  been  observed  to  have  short  and  ets  have  nc 
feeble  tails,  and  not  a  few  have  been  entirely  without  them.     Those  tails- 
of  1585  and  1763  offered  no  vestige  of  a  tail ;  and  Cassini  describes  the 
comet  of  1682  as  being  as  round  and  as  bright  us  Jupiter.     On  the  other 
hand,  instances  are  not  wanting  of  comets  furnished  with  many  tails, 
or  streams  of  diverging  light.     That  of  1744  had  no  less  than  six, 
spread  out  like  an  immense  fan,  extending  to  a  distance  of  nearly  30 
degrees  in  length. 

"  The  smaller  comets,  such  as  are  visible  only  in  telescopes,  or  with 
difficulty  bv  the  naked  eye,  and  which  are  by  far  the  most  numerous, 
offer  very  frequently  no  appearance  of  a  tail,  and  appear  only  as  round 
or  somewhat  oval  vaporous  masses,  more  dense  toward  the  center; 
where,  however,  they  appear  to  have  no  distinct  nucleus,  or  anything 
which  seems  entitled  to  be  considered  as  a  solid  body. 

"  The  tail  of  the  comet  of  1456  was  60  degrees  long.     That  of  1618,    others  hav« 
100  degrees,  so  that  its  tail  had  not  all  risen  when  its  head  reached  the  several  taiU 
middle  of  the  heavens.     The  comet  of  1680  was  so  great,  that  though 
its  head  set  soon  after  the  sun,  its  tail,  70  degrees  long,  continued  visi- 
ble all  night.     The  comet  of  1689  had  a  tail  66  degrees  long.     That  of 
1769  had  a  tail  more  than  90  degrees  in  length.     That  of  1811  had  a 
tail  23  degrees  long.     The  recent  comet  of  1843  had  a  tail  60  degrees 
in  length." 

The  following  figure  gives  a  telescopic  view  of  the  comet  of  1811. 


.58  ASTRONOMY. 

CHAP.  XII.  "  When  w«  have  determined  the  elements  of  a  comet's  orbit,  we  com- 
~~~tg  Pare  them  with  those  of  comets  before  observed,  and  see  whether  there 
of  civets  is  an  agreement  with  respect  to  any  of  them.  If  there  is  a  perfect 
how  deter-  identity  as  to  the  elements,  we  should  have  no  hesitation  in  concluding 
mined.  that  they  belonged  to  different  appearances  of  the  same  comet.  But 

this  condition  is  not  rigorously  necessary  ;  for  the  elements  of  the 
orbit  may,  like  those  of  other  heavenly  bodies,  have  undergone  changes 
from  the  perturbations  of  the  planets  or  their  mutual  attractions.  Con- 
sequently, we  have  only  to  see  whether  the  actual  elements  are  nearly 
the  same  with  those  of  any  comet  before  observed,  and  then,  by  the  doc- 
trine of  chances,  we  can  judge  what  reliance  is  to  be  placed  upon  this 

resemblance." 

Comet  of  1811 


Dr.Halley'i      "Dr.  Halley  remarked  that  the  comets  observed  in  1531,  1607,  1682, 
prediction       na(j  nearly  the  same  elements  ;  and  he  hence  concluded  that  they  be- 
rerified.         longed  to  the  same  comet,  which,  in  151  years,  made  two  revolutions, 
its  period  being  about  76  years.     It  actually  appeared  in  1759,  agreea- 
bly to  the  prediction  of  this  great  astronomer  ;  and  again  in  1832,  by 
the  computation  of  several  eminent  astronomers.     According  to  Kep- 
ler's third  law,  if  we  take  for  unity  half  the  major  axis  of  the  earth's 
Particulars  orbit,  the  mean  distance  of  this  comet  must  be  equal  to  the  cube  root 
of  comets.      of  the  square  of  76,  that  is,  to  17.95.     The  major  axis  of  its  orbit  must, 
therefore,  be  35.9  ;  and  as  its  observed  perihelion  distance  is  found  t:> 
ba  0.58.   it  follows   that   itn  aphelion  distance  is  equal  to  35.32.     It 


COMETS.  159 

departs,  therefore,  from  the  sun  to  thirty-five  times  the  distance  of  the  CHAP.  XH 
earth,  and  afterward  approaches  nearly  twice  as  near  the  sun  as  the 
earth  is,  thus  describing  an  ellipse  extremely  elongated. 

"The  intervals  of  its  return  to  its  perihelion  are  not  constantly  the 
same.  That  between  1531  and  1607  was  three  months  longer  than 
that  between  1607  and  1682  ;  and  this  last  was  18  months  shorter  than 
the  one  between  1682  and  1759.  It  appears,  therefore,  that  the  motions 
of  comets  are  subject  to  perturbations,  like  those  of  the  planets,  and  to 
a  much  more  sensible  degree. 

"  Elements  of  the  Orbits  of  the  three,  Comets,  which  have  appeared  ac- 
cording to  prediction,  taken  from  the  work  of  Professor  Littrow. 

Halley.       Encke.        Biela. 

Longitude  of  the  ascending  node,     -          54C          335°       249° 
Inclination  of  the  orbit  to  the  ecliptic,    162°  13°         13° 

Longitude  of  the  perihelion,     -         -        303°         157°        108° 
Greatest semidiameter,  that  of  the  earth)     jg  QQ  Qg 

being  called  1,  -         -         -         -         ) 

Least  semidiametqf.          -         -         -  4.6  1.2  2.4 

Time  of  revolution  in  years,          -  76  3.29         6.74 

Nov.  16.         May  4.      Nov.  27 

Time  of  the  perihelion  passage,     -  1835          1832         1832 

"  The  comets  of  Encke  and  Biela  move  according  to  the  order  of  thr> 
signs  of  the  zodiac,  or  have  their  motions  direct;  the  motion  of  that 
of  Halley  is  retrograde. 

"Comets,  in  passing  among  and   near   the  planets,  are  materially          Jupitei, 
drawn  aside  from  their  courses,  and  in  some  cases  have  their  orbits  en-  andhissatel- 
tirely  changed.     This  is  remarkably  the  case  with  Jupiter,  which  seems,  lltes>  a  gieat 
by  some  strange  fatality,  to  be  constantly  in  their  way,  and  to  serve  as  *    ck        h 
a  perpetual  stumbling-block  to  them.     In  the  case  of  the  remarkable  comets 
comet  of  1770,  which  was  found  by  Lexell  to  revolve  in  a  moderate 
ellipse  in  the  period  of  about  five  years,  and  whose  return  was  pre- 
dicted by  him  accordingly,  the  prediction  was  disappointed  by  the  comet 
actually  getting  entangled  among  the  satellites  of  Jupiter,  and  being 
completely  thrown  out  of  its  orbit  by  the  attraction  of  that  planet,  and 
forced  into  a  much  larger  ellipse.     By  this  extraordinary  renconter, 
the  motions  of  the  satellites  suffered  not  the  least  perceptible  derangement — 
a  sufficient  proof  of  the  smallness  of  the  comet's  mass." 

The  comet  of  1456,  represented  as  having  a  tail  of  60°  in  length,  is 
now  found  to  be  Halley's  comet,  which  has  made  several  returns  — 
in  1531,  1607,  1682,  1759, and  recently,  in  1835.  In  1607  the  tail  was 
said  to  have  been  over  30°  in  length  ;  but  in  1835  the  tail  did  not  ex- 
ceed 12°  Does  it  lose  substance,  or  does  the  matter  composing  the 
tail  condense  ?  or,  have  we  received  only  exaggerated  and  distorted 
accounts  from  the  earlier  times,  such  as  fear,  superstition,  and  uwe, 
always  put  forth  ?  We  ask  these  questions,  but  cannot  answer  them 


1  eO  A  S  T  R  0  N  O  M  Y. 

CHAT.  XII.       The   following  cut    represents   the   appearance   of    the   comet   of 
1819. 


Fears  en-      "Professor  Kendall,  in  his  Uranography,  speaking  of  the  fears  occa- 

tertained,  by  gjoned  by  comets,  says:  "Another  source  of  apprehension,  with  regard 

•ome,     that  ^Q  comejSj  ar}ses  from  the  possibility  of  their  striking  our  earth.     It  is 

may  quite  probable  that  even  in  the  historical  period  the  earth  has  been 

come      into  enveloped  in  the  tail  of  a  comet.     It  is  not  likely  that  the  effect  would 

collision  with  be  sensible  at  the  time.     The  actual  shock  of  the  head  of  a  comet  against 

our  earth.       the  earth  is  extremely  improbable.     It  is  not  likely  to  happen  once  in 

a  million  of  years. 

"  If  such  a  shock  should  occur,  the  consequences  might  perhaps  be 
very  trivial.  It  is  quite  possible  that  many  of  the  comets  are  not 
heavier  than  a  single  mountain  on  the  surface  of  the  earth.  It  is  w^ll 
known  that  the  size  of  mountains  on  the  earth  is  illustrated  by  com- 
paring them  to  particles  of  dust  on  a  common  globe." 


CHAPTER    XIII. 

ON    THE   PECULIARITIES   OF   THE   FIXED    STARS. 

CHAP,  xin.  pOR  the  factg  ag  contained  in  the  subject  matter  of  this 
chapter,  we  must  depend  wholly  on  authority ;  for  that  reason 
we  give  only  a  compilation,  made  in  as  brief  a  manner  as  the 
nature  of  the  subject  will  admit. 

In  the  first  part  of  this  work  it  was  soon  discovered  that 
the  fixed  stars  were  more  remote  than  the  sun  or  planets ; 
and  now,  having  determined  their  distances,  we  may  make 
further  inquiries  as  to  the  distances  to  the  stars,  which  will 


FIXED  STARS.  161 

give  some  index  by  which  to  judge  of  their  magnitudes,  nature,  CHAP.  XIIL 
and  peculiarities. 

"  It  would  be  idle  to  inquire  whether  the  fixed  stars  have  a  sensible     Ba«e  from 
parallax,  when  observed  from  different  parts  of  the  earth.     We  have  wnich        *o 
already  had  abundant  evidence  that  their  distance  is  almost  infinite.     It  mca§ur"    to 
is  only  by  taking  the  longest  base  accessible  to  us,  that  we  can  hope  to 
arrive  at  any  satisfactory  result. 

"Accordingly,  we  employ  the  major  axis  of  the  earth's  orbit,  which  is 
nearly  200  millions  of  miles  in  extent.  By  observing  a  star  from  the 
two  extremities  of  this  axis,  at  intervals  of  six  months,  and  applying  a 
correction  for  all  the  small  inequalities,  the  effect  of  which  we  have 
calculated,  we  shall  know  whether  the  longitude  and  latitude  are  the 
came  or  not  at  these  two  epochs. 

"  It  is  obvious,  indeed,  that  the  star  must  appear  more  elevated  above          Annual 
the  plane  of  the  ecliptic  when  the  earth  is  in  the  part  of  its  orbit  which  parallax, 
is  nearest  to  the  star,  and  more  depressed  when  the  contrary  takes 
place.     The  visual  rays  drawn  from  the  earth  to  the  star,  in  these  ivvG 
positions,  differ  from  the  straight  line  drawn  from  the  star  to  the  center 
of  the  earth's  orbit ;  and  the  angle  which  either  of  them  forms  with 
this  straight  line,  i*  called  the  annual  parallax. 

"  As  the  earth  does  not  pass  suddenly  from  one  point  of  its  orbit  to     The  effect 
the  opposite,  but  proceeds  gradually,  if  we  observe  the  positions  of  a  °f  a  »«n»iuJ» 
star  at  the  intermediate  epochs,  we  ought,  if  the  annual  parallax  is  sen-  para  ax 
sible,  to  see  its  effects  developed  in  the  same  gradual  manner.     For 
example,  if  the  star  is  placed  at  the  pole  of  the  ecliptic,  the  visual  rays 
drawn  irom  it  to  the  earth,  will  form  a  conical  surface,  having  its  apex 
at  the  star,  and  for  its  base,  the  earth's  orbit.     This  conical  surface 
being  produced  beyond  the  star,  will  form  another  opposite  to  the  first, 
and  the  intersection  of  this  last  with  the  celestial  sphere,  will  constitute 
a  small  ellipse,  in  which  the  star  will  always  appear  diametrically  oppo- 
site to  the  earth,  and  in  the  prolongation  of  the  visual  rays  drawn  to 
the  apex  of  the  cones. 

"But  notwithstanding  all  the  pains  that  have  been  taken  to  multiply    The  annual 
observations,  and  all  the  care  that  has  been  used  to  render  them  per-  parallaxmnst 
fectly  exact,  we  have  been  able  to  discover  nothing  which  indicates,  ^  leBS  lhan 
with  certainty,  even  the  existence  of  an  annual  parallax,  to  say  nothing  on 
of  its  magnitude.     Yet  the  precision  of  modern  observations  is  such, 
that  if  this  parallax  were  only  1",  it  is  altogether  probable  that  it  would 
not  have  escaped  the  multiplied  efforts  of  observers,  and  especially  those 
of  Dr.  Bradley,  who  made  many  observations  to  discover  it,  and  who, 
in  this  undertaking,  fell  unexpectedly  upon  the  phenomena  of  aberra- 
tion* and  nutation.     These  admirable  discoveries  have  themselves 
served  to  show,  by  the  perfect  agreement  which  is  thus  found  to  take 

•  Subject  to  be  explained  hereafter. 


162  ASTRONOMY. 

CHAP.  XIII.  place  among  observations,  that  it  is  hardly  to  be  supposed  that  the 
annual  parallax  can  amount  to  1".  The  numerous  observations  of  the 
pole  star,  recently  employed  in  measuring  an  arc  of  the  meridian 
through  France,  have  been  attended  with  a  similar  result,  as  to  the 
amount  of  the  annual  parallax.  From  all  this  we  may  conclude,  that 
as  yet  there  are  strong  reasons  for  believing  that  the  annual  parallax 
is  less  than  1",  at  least  with  respect  to  the  stars  hitherto  observed. 

"  Thus  the  semidiameter  of  the  earth's  orbit,  seen  from  the  nearest 

star,  would  not  appear  to  subtend  an  angle  of  1'  ;  and  to  an  observer 

placed  at  this  distance, our  sun,  with  the  whole  planetary  system,  would 

occupy  a  space  scarcely  exceeding  the  thickness  of  a  spider's  thread. 

Conclusion      "  If  these  results  do  not  make  known  the  distance  of  the  stars  from 

to  be  drawn  tne  earth,  the^  at  least  teach  us  the  limit  beyond  which  the  stars  must 
86  necessarily  be  situated.  If  we  conceive  a  right-angled  triangle,  having 
for  its  base  half  the  major  axis  of  the  earth's  orbit,  and  for  its  vertex 
an  angle  of  1  ,  the  distance  of  this  vertex  from  the  earth,  or  the  length 
of  the  visual  ray,  will  be  expressed  by  212207,  the  radius  of  the  earth's 
orbit  being  unity ;  and  as  this  radius  contains  23987  times  the  semidia- 
meter of  the  earth,  it  follows  that  if  the  annual  parallax  of  a  star  were 
only  1",  its  distance  from  the  earth  would  be  equal  to  5090209309  radii 
of  the  earth,  or  20086868036404  miles ;  that  is,  more  than  20  billions. 
But  if  the  annual  parallax  is  less  than  1",  the  stars  are  beyond  the  limit 
which  we  have  assigned. 
Changes  "  It  is  evident  that  the  stars  undergo  considerable  changes,  since  these 

(n  individual  changes  are  sensible  even  at  the  distance  at  which  we  are  placed.  There 

******  are  some  which  gradually  lose  their  light,  as  the  star  (T  of  Ursa  Major. 

Others,  as  /fi  of  Cetus,  become  more  brilliant.  Finally,  there  are  some 
which  have  been  observed  to  assume  suddenly  a  new  splendor,  and  then 
gradually  fade  away.  Such  was  the  new  star  which  appeared  in  1572, 
A  new  star,  in  the  constellation  Cassiopeia.  It  became  all  at  once  so  brilliant  that 
it  surpassed  the  brightest  stars,  and  even  Venus  and  Jupiter  when 
nearest  the  earth.  It  could  be  seen  at  midday.  Gradually  this  great 
brilliancy  began  to  diminish,  and  the  star  disappeared  in  sixteen  months 
from  the  time  it  was  first  seen,  without  having  changed  its  place  in  the 
heavens.  Its  color,  during  this  time,  suffered  great  variations.  At  first 
it  was  of  a  dazzling  white,  like  Venus  ;  then  of  a  reddish  yellow,  like 
Mars  and  Aldebaran  ;  and  lastly,  of  a  leaden  white,  like  Saturn.  An- 
Another  otner  star  which  appeared  suddenly  in  1604,  in  the  constellation  Ser- 

n«w  star.  pentarias,  presented  similar  variations,  and  disappeared  after  several 
months.  These  phenomena  seem  to  indicate  vast  flames  which  burst 
forth  suddenly  in  these  great  bodies.  Who  knows  that  our  sun  may 
not  be  subject  to  similar  changes,  by  which  great  revolutions  have 
perhaps  taken  place  in  the  state  of  our  globe,  and  are  yet  to  take  place. 
Periodical  •«  Some  stars,  without  entirely  disappearing,  exhibit  variations  not  less 
remarkable.  Their  light  increases  and  decreases  alternately  in  regular 
periods.  They  are  called  for  this  reason  variable  stars.  Such  is  the 


FIXED  STARS.  163 

star  Algol,  in  the  head  of  Medusa,  which  has  a  period  of  about  three  CHAP.  XIIL 
days  ;  3  of  Cepheus,  which  has  one  of  five  days  ;  @  of  Lyra,  six  ;  ft  of 
Antinous,  seven  ;  o  of  Cetus,  334  ;  and  many  others. 

"Several  attempts  have  been  made  to  explain  these  periodical  varia-       Attempt* 
tions.     It  is  supposed  that  the  stars  which  are  subject  to  them,  are,  like  to      explain 
all  the  other  stars,  self-luminous  bodies,  or  true  suns,  turning  on  their  periodical 
ax'is,  and  having  their  surfaces  partly  covered  with  dark  spots,  which  chanSes- 
may  be  supposed  to  present  themselves  to  us  at  certain  times  only,  in 
consequence  of  their  rotation.     Other  astronomers  have  attempted  to 
account  for  the  facts  under  consideration,  by  supposing  these  stars  to 
have  a  form  extremely  oblate,  by  which  a  great  difference  would  take 
place  in  the  light  emitted  by  them  under  different  aspects.     Lastly,  it 
has  been  supposed  that  the  effect  in  question  is  owing  to  large  opake 
bodies,  revolving  about  these  stars,  and  occasionally  intercepting  a  part 
of  their  light.     Time  and  the  multiplication  of  observations  may  per- 
haps decide  which  of  these  hypotheses  is  the  true  one. 

"  One  of  the  best  methods  of  observing  these  phenomena  is  to  compare        Order  in 
the  stars  together,  designating  them  by  letters  or  numbers,  and  dispos-  these  obsw- 
ing  them  in  the  order  of  their  brilliancy.     If  we  find,  by  observation,  varans, 
that  this  order  changes,  it  is  a  proof  that  one  of  the  stars  thus  com- 
pared, has  likewise  changed  ;  and  a  few  trials  of  this  kind  will  enable  us 
to  ascertain  which  it  is  that  has  undergone  a  variation.     In  this  man- 
ner, we  can  only  compare  each  star  with  those  which  are  in  the  neigh- 
borhood, and  visible  at  the  same  time.     But  by  afterward  comparing 
these  with  others,  we  can,  by  a  series  of  intermediate  terms,  connect 
together  the  most  distant  extremes.     This  method,  which  is  now  prac- 
ticed, is  far  preferable  to  that  of  the  ancient  astronomers,  who  classed 
the  stars  after  a  very  vague  comparison,  according  to  what  they  called 
the  order  of  their  magnitudes,  but  which  was,  in  reality,  nothing  but 
that  of  their  brightness,  estimated  in  a  very  imperfect  manner. 

"  By  comparing  the  places  of  some  of  the  fixed  stars,  as  determined  Suggestion 
from  ancient  and  modern  observations,  Dr.  Halley  discovered  that  they  ofDr.Halley. 
had  a  proper  motion,  which  could  not  arise  from  parallax,  precession, 
or  aberration.  This  remarkable  circumstance  was  afterward  noticed 
by  Cassini  and  Le  Monnier,  and  was  completely  confirmed  by  Tobias 
Mayer,  who  compared  the  places  of  80  stars,  as  determined  by  Roemer, 
with  his  own  observations,  and  found  that  the  greater  part  of  them 
had  a  proper  motion.  He  suggested  that  the  change  of  place  might 
arise  from  a  progressive  motion  of  the  sun  toward  one  quarter  of  the 
heavens  ;  but  as  the  result  of  his  observation  did  not  accord  with  his 
theory,  he  remarks  that  many  centuries  mirst  elapse  before  the  true 
cause  of  this  motion  could  be  explained. 

"  The  probability  of  a  progressive  motion  of  the  sun  was  suggested 
upon  theoretical  principles  by  the  late  Dr.  Wilson  of  Glasgow  ;  and 
Lalande  deduced  a  similar  opinion  from  the  rotatory  motion  of  the  sun, 
by  supposing,  that  the  same  mechanical  force  which  gives  it  a  motion 


164  ASTRONOMY. 

CHAP.  XIII.  round  its  axis,  would  also  displace  its  center,  and  give  it  a  motion  of 

translation  in  absolute  space 

ns""  "  If  the  sun  has  a  motion  in  absolute  space,  directed  toward  any 
such  a  the-  quarter  °f  tne  heavens,  it  is  obvious  that  the  stars  in  that  quarter  must 
OTV-  appear  to  recede  from  each  other,  while  those  in  the  opposite  region 

would  seem  gradually  to  approach,  in  the  same  manner  as  when  walk- 
ing through  a  forest,  the  trees  toward  which  we  advance  are  constantly 
separating,  while  the  distance  of  those  which  we  leave  behind  is  gradu- 
ally contracting.  The  proper  motion  of  the  stars,  therefore,  in  opposite 
regions,  as  ascertained  by  a  comparison  of  ancient  with  modern  obser- 
vations, ought  to  correspond  with  this  hypothesis  ;  and  Sir  W.  Her- 
schel  found,  that  the  greater  part  of  them  are  nearly  in  the  direction 
which  would  result  from  a  motion  of  the  sun  toward  the  constellation 
Hercules,  or  rather  to  a  part  of  the  heavens  whose  right  ascension  is 
250°  52'  30",  and  whose  north  polar  distance  is  40°  22'.  Klugel  found 
the  right  ascension  of  this  point  to  be  260°,  and  Prevost  made  it  230°, 
with  65°  of  north  polar  distance.  Sir  W.  Herschel  supposes  that  the 
motion  of  the  sun,  and  the  solar  system,  is  not  slower  than  that  of  the 
earth  in  its  orbit,  and  that  it  is  performed  round  some  distant  center. 
The  attractive  force  capable  of  producing  such  an  effect,  he  does  no< 
suppose  to  be  lodged  in  one  large  body,  but  in  the  center  of  gravity  of 
a  cluster  of  stars,  or  the  common  center  of  gravity  of  several  clusters." 
The  following  figures,  taken  from  Norton's  Astronomy,  represent 
the  telescopic  appearance  of  some  of  the  double  stars. 

Doable  "  There  are  stars  which,  when  viewed  by  the  naked  eye,  and  even 
and  multiple  by  the  help  of  a  telescope  of  moderate  power,  have  the  appearance  of 
•tan.  on]v  a  sing|e  star  ;  but,  being  seen  through  a  good  telescope,  they  are 

found  to  be  double,  and  in  some  cases  a  very  marked  difference  is  per- 
ceptible, both  as  to  their  brilliancy  and  the  color  of  their  light.  These 
Sir  W.  Herschel  suppos/d  to  be  so  near  each  other,  as  to  obey  recipro* 
cally  the  power  of  each  other's  attraction,  revolving  about  their  com- 
mon center  of  gravity,  in  certain  determinate  periods. 


Castor,     y  Leonis,          Rigel,       Pole  Star,    jrMonoc,    gCancri. 

Revolutions      "  The  two  stars,  for  example,  which  form  the  double  star  Castoi 

of  the  multi-  have  varied  in  their  angular  situation  more  than  45°  since  they  were 

pie  stars.        observed  by  Dr.  Bradley,  in  17595  and  appear  to  perform  a  retrograde 

revolution  in  342  years,  in  a  plane  perpendicular  to  the  direction  of  the 

sun.     Sir  W.  Herschel  found  them  in  intermediate  angular  positions, 

at  intermediate  times,  but  never  could  perceive  any  change  in  their 

distance.     The  retrograde  revolution  of  y  in  Leo,  another  double  star, 

is  supposed  to  be  in  a  plane  considerably  inclined  to  the  line  in  which 

we  view  it,  and  to  be  completed  in  1200  years.     The  stars  <  of  Bootes, 


FIXED    STARS.  165 

perform  a  direct  revolution  in  1681  years,  in  a  plane  oblique  to  the  sun.  CHAP.  Yrq, 
The  stars  £  of  Serpens,  perform  a  retrograde  revolution  in  about  375 
years;  and  those  of  y  in  Virgo  in  708  years,  without  any  change  of 
their  distance.  In  1802,  the  large  star  £  of  Hercules,  eclipsed  the 
smaller  one,  though  they  were  separate  in  1782.  Other  stars  are  sup- 
posed to  be  united  in  triple,  quadruple,  and  still  more  complicated 
systems. 

"With  respect  to  the  determination  of  the  real  magnitude  of  the  stars,    Descriptioa 
and  their  respective  distances,  we  have  as  yet  made  but  little  progress,  of  nebula. 
Researches  of  this  kind  must  be  left  to  future  astronomers.     It  appears, 
however,  that  the   stars  are   not  uniformly  distributed  through   the 
heavens,  but  collected  into  groups,  each  containing  many  millions  of 
stars.     We  can  form  some  idea  of  them  from  those  small  whitish  spots 
called  Nebulae,  which  appear  in  the  heavens  as  represented  in  the  ac- 
companying illustration.    By  means  of  the  telescope,  we  distinguish  in 
these  collections  an  almost  infinite  number  of  small  stars,  so  near  each 
Other,  that  their 
rays  are  ordina- 
rily blended  by 
irradiation,  and 
thus  present  to 
the  eye  only  a 
faint     uniform 
sheet  of    light. 
That     large, 
white,    lumi- 
nous    track, 
which  traverses 
the    heavens 
from  one  pole  to 
the  other,  under 

the  name  of  the  Milky  Way,  is  probably  nothing  but  a  nebula  of  th*s     The  Milk} 
kind,  which  appears  larger  than  the  others,  because  it  is  nearer  to  u*.  Way  a  ne 
With  the  aid  of  the  telescope  we  discover  in  this  zone  of  light  such  a  bula' 
prodigious   number  of  stars   that   the    imagination  is  bewildered   in 
attempting  to  represent  them.     Yet  from  the  angular  distances  of 
these  stars,  it  is  certain  that  the  space  which  separates  those  which 
seem  nearest  to  each  other,  is  at  least  a  hundred  thousand  times  as  great 
as  the  radius  of  the  earth's  orbit.     This  will  give  us  some  idea  of  the 
immense  t-xtent  of  the  group.     To  what  distance  then  must  we  with- 
draw, in  order  that  this  whole  collection  may  appear  as  small  as  the 
other  nebulae  which  we  perceive,  some  of  which  cannot,  by  the  assist- 
ance of  the  best  telescopes,  be  made  to  present  anything  but  a  bright 
speck,  or  a  simple  mass  of  light,  of  the  nature  of  which  we  are  able  to 
form  some  idea  only  by  analogy  ?    When  w«  pttetnpt,  in  imagination, 
to  fathom  this  abyss,  it  is  in  vain  to  think  of  prescribing  any  limits  to 


166  ASTRONOMY. 

CHAP.  XIiI.   the  universe,  and  the  mind  reverts  involuntarily  to  the  insignificant 

portion  of  it  which  we  are  destined  to  occupy. " 

Observa-  Before  we  close  this  chapter,  we  think  it  important  to  call  the  atten- 
tions on  ta-  t|on  Of  tne  reader  to  table  II  ,  in  which  will  be  seen,  at  a  glance  (in 
the  columns  marked  annual  variation),  the  general  effect  of  the  preces- 
sion of  the  equinoxes  ;  and  although  we  have  called  particular  attention 
to  the  fact  elsewhere,  we  here  notice  that  all  the  stars,  from  the  6th  to 
the  3£th  hour  of  right  ascension,  have  a  progressive  motion  to 
the  southward  ( — ),  and  all  the  stars  from  the  18th  to  the  6th  honr 
of  right  ascension  have  a  progressive  motion  to  the  northward  (-)-),  and 
the  greatest  variations  are  at  0  h.  and  12  h.  But  these  motions  are  not,  in 
reality,  the  motions  of  the  stars  ;  they  result  from  motions  of  the  earth. 
Whenever  the  annual  motion  of  any  star  does  not  correspond  with  this 
common  displacement  of  the  equinox,  we  say  the  star  has  a  proper 
motion  ;  and  by  such  discrepancy  it  has  been  decided,  that  those  stars 
marked  with  an  asterisk,  in  the  catalogue,  have  proper  motions  ;  and 
the  star  61  Cygni,  near  the  close  of  the  table,  has  the  greatest  proper 
motion. 

The  paral-      From  this  circumstance,  and  from  the  fact  of  its  being  a  double  star, 

i&x    of    61  it  was  selected  by  Bessel  as  a  fit  subject  for  the  investigation  of  stellar 

Cygni  disco-  paranax  ;  an(j  it  is  now  contended,  and  in  a  measure  granted,  that  the 

annual  parallax  of  this  star  is  0".35,  which  makes  its  distance  more 

than  592.000  times  the  radius  of  the  earth's  orbit ;  a  distance  that  light 

could  not  traverse  in  less  than  nine  and  one-fourth  years. 


PHYSICAL  ASTRONOMY.  16? 

SECTION   III. 
PHYSICAL   ASTRONOMY. 

CHAPTER  I. 

GENERAL   LAWS   OF   MOTION THE   THEORY   OF   GRAVITY. 

CHAP.  1. 

(148.)  IN  a  work  like  this,   designed  for  elementary  in-  mat  &ho~ld 
struction,  it  cannot  be  expected  that  a  full  investigation  of  be  exPecte*» 
physical  astronomy  shall  be  entered  into ;  for  that  subject  »n  this  work, 
alone  would  require  volumes ;  and  to   fully  appreciate  and 
comprehend   it,  requires  the  matured  philosopher  combined 
with  the  accomplished  mathematician. 

We  shall  give,  however,  a  sufficient  amount  to  impart  a  good 
general  idea  of  the  subject  —  if  one  or  two  points  are  taken 
on  trust. 

For  elementary  principles  we  must  turn  a  moment  to  natu-    Elemental 
ral  philosophy,  and  consider  the  laws  of  inertia,  motion,  and  principles. 
force.    Motion  is  a  change  of  place  in  relation  to  other  bodies 
which  we  conceive  to  be  at  rest ;  and  the  extent  of  change  in 
the  time  taken  for  unity  is  called  velocity,   and  the  essential 
cause  of  motion  we  denominate  force. 

A  double  force  will  give  a  double  velocity  to  bodies  moving  Velocity  the 

/•        i       •  •  i  •  ••  i  •  *    *    ?     measure      of 

freely  in  void  space,  or  in  an  unresisting  medium  —  a  tnple  force 
force,  a  triple  velocity,  &c.     This  is  taken  as  an  axiom  — and 
hence,  when  we  consider  mere  material  points  in  motion,  the 
relative  velocities  measure  the  relative  amounts  of  force. 

There  are  three  elements  to  motion,  which  the  philosopher 
never  loses  sight  of;  or  we  may  say  that  he  never  thinks  of 
motion  without  the  three  distinct  elements  of  time,  velocity,  and 
distance,  coming  into  his  mind. 

Algebraically,  we  put  t,  v,  and  d,  to  represent  the  three  ele- 
ments, and  then  we  have  this  important  and  general  equation, 
tv  =  d  (1) 


168  ASTRONOMY. 

CHAP.  1.  d  d 

From  this  we  derive  v=—  (2)  and  *=-  C3) 

Expression  t  V  ^    ' 

tv  force 

(  149.)  As  forces  are  in  proportion  to  velocities  (when  mo- 
mentum is  not  in  question ),  therefore,  if  we  put  /  and  F  to 
represent  two  forces  corresponding  to  the  distances  d  and  D, 
which  are  described  in  the  times  t  and  T,  then  by  making  use 
of  equation  (  2  ),  in  place  of  the  velocities,  we  have 

f  •  F  ••  -  i  —  (  4  ^  * 

w    *  t         T  v/ 

The  law  of  (  150.  )  A  body  at  rest,  has  no  power  to  put  itself  in  mo- 
tion ;  and  having  no  self  power,  no  internal  force  or  will,  m 
any  shape,  it  cannot  increase  or  diminish  the  motion  it  may 
have,  or  change  the  direction  it  may  be  moving.  This  is  the 
law  of  inertia.  It  cannot  of  itself  change  its  state  ;  and  if  it 
is  changed  it  must  be  acted  upon  by  some  external  force ; 
and  this  accords  with  universal  experience ;  and  this  law  is 
the  most  natural  and  simple  of  any  we  can  imagine,  but  it  is 
only  in  the  motion  of  the  heavenly  bodies  that  it  is  fully 
exemplified. 

Bwne  central      The    earth,  moon,  and  planets   move  in  curves  —  not  in 
force     mnst  j-jg]^   lmes.      The  directions  of  their  motions  are  changed. 
motions    of  Something  external  from  them  must,  therefore,  change  them ; 
the     earth,  for  fae  law  Of  inertia  would  continue  a  motion  once  obtained 
planets.        iQ  a  straight  line.     Now  this  force  must  exist  within  the  or- 
bit of  every  curve;  we  therefore  naturally  refer  it  to  the 
body  round  which  others  circulate.     The  earth  and  planets 
go  round  the  sun,  and  if  we  could  suppose  a  force  residing  in 
the  sun  to  extend  throughout  the  system  sufficient  to  draw 
bodies  to  it,   this  would  at  once  account  not  only  for  the 
planets  deviating  from  a  right  line,  but  would  account  for  a 
constant  deviation  of  all  bodies  to  that  point,  and  the  preser- 
vation  of  the  system. 

Th«  moon's      The  moon  goes  round  the  earth,  constantly  deviating  from 
Motion  con.  ^  tangent  of  its  orbit,  and  the  law  of  inertia  is  constantly 


sidered. 


»  We  number  the  proportions  the  same  aa  equations,  for  a  propor- 
tion is  but  an  equation  in  another  form. 


THE  EARTH'S  ATTRACTION.  169 

urging  it  to  rise  from  the  center ;  the  two  on  an  average  balan-    CHAP.  I. 
cing  each   other,  retains  the  inoou   in  an  orbit  about  the 
earth. 

Now  what  and  where  is  this  force  ?  Is  it  around  the 
earth,  or  within  the  earth  ?  Is  it  electrical  or  magnetic  ?  or 
is  it  that  same  force  (  call  it  what  we  may )  that  makes  a 
body  fall  toward  the  earth's  center  when  unsupported  on  a 
resting  base  ? 

A  trifling  incident,  the  fall  of  an  apple  from  a  tree,  seems    contempia. 
to  have  led  the  mind  of  Newton  to  the  contemplation  of  this  Hons  of  SH 
force  which  compels  and  causes  bodies  to  fall,  and  he  at  once  ^ac   Ne> 
conceived  this  force  to  extend  to  the  moon  and  to  cause  it  to 
deviate  from  the  tangent  of  its  orbit. 

The  next  consideration  was,  whether  if  this  were  the  force, 
it  was  the  same  at  the  distance  of  the  moon,  as  on  the  sur- 
face of  the  earth  ;  or  if  it  extended  with  a  diminished  amount, 
what  was  the  law  of  diminution  ? 

Newton  now  resorted  to  computation,  and  for  a  test  he       incipient 
conceived  the  force  in  question  to  extend  to  the  moon,  undi-  steps  to  ^ 
minished  by  the  distance ;  and  corresponding  thereto  he  de-  gravity, 
cided  that  the  moon  must  then  make  a  revolution  in  its  orbit 
in  10  h.  55m.      But  the  actual  time  is  27  d.  7h.   43m., 
which  shows  that  if  the  force  is  the  same  which  pervades  a 
falling  body  on  the  surface  of  the  earth,  it  must  be  greatly 
diminished. 

Now  by  making  a  reverse  computation,  taking  the  actual     important 
time  of  revolution,  and  finding  how  far  the  moon  did  really  comPuta- 
fall  from  the  tangent  of  its  orbit  in  one  second  of  time,  it  was 
found  to  be  about  ^^  part  of  16  T^  feet  —  the  distance  a 
body  falls  the  first  second  of  time. 

But  the  distance  to  the  moon  is  about  60  times  the  radius 
of  the  earth,  and  the  inverse  square  of  this  is  ^g-V^*  which 
corresponds  to  the  actual  fall  of  the  moon  in  one  second. 

(151.)  It  is   a  well-established  fact  in  philosophy,  and    A  principle 
geometrically  demonstrated,  that  any  force  or  influence  exist-  'BPhllosoP°* 
ing  at  a  point,  must  dimmish  as  it  spreads  over  a  larger 
space,  and  in  proportion  to  the  increase  of  space.    But  space 
increases  as  the  square  of  linear  distance,  as  we  see  by  Fig.  28, 


170  ASTRONOMY. 

CHAP,  i.         A  double  distance  spreads  the  influence  over  four  times  the 

space,  whatever  that  influence  may  be;  a  triple  distance,  nine 

times  the  space,  etc.,  the  space  increasing  as  the  square  of 

Fig.  28. 


the  distance.  Therefore,  any  influence  spreading  in  all  di- 
rections from  its  central  point  must  be  enfeebled  as  the  square 
of  the  distance. 

The  theory      From  observations  and  considerations  like  these,  Newton 
gravity.        established  the  all-important  and  now  universally  admitted 
theory  of  gravity. 

This  theory  may  be  summarily  stated  in  the  following 
words : 

Every  body  of  matter  in  the  universe  attracts  every  other  body, 
in  direct  proportion  to  its  mass,  and  in  the  inverse  proportion  to 
the  square  of  the  distance. 

This  theory      Some  attempts  have  been  made,  from  time  to  time,  to  call 
wen    estab-  fae  ^ruth  of  this  theory  in  question,  and  substitute  in  its 
place  the  influence  of  light,  caloric,  and  electricity;  but  any 
thing  like  a  close  application  shows  how  feebly  all  such  sub- 
stitutes stand  the  test. 

The  theory  of  gravity  so  exactly  accounts  for  all  the  phy- 
sical phenomena  of  the  solar  system,  that  it  is  impossible  it 
should  be  false ;  and  although  we  cannot  determine  its  nature 
or  its  essence,  it  is  as  unreasonable  to  doubt  its  existence,  as 
to  doubt  the  existence  of  animate  beings,  because  we  know 
nothing  of  the  principle  of  life. 

Attraction  (152.)  According  to  the  theory  of  gravity,  every  particle 
*f  *^!"esu*  composing  a  body  has  its  influence,  and  a  very  irregular  body 
may  be  divided  in  imagination  into  many  smaller  bodies,  and 
the  center  of  gravity  of  each  taken  as  the  point  of  attraction, 
and  all  the  forces  resolved  into  one  will  be  the  attraction  of 
thf  whole  body. 


STANDARD    OF    FORCE.  171 

In  a  sphere  composed  of  homogeneous  particles,  the  aggre-   CHAP.  L 
gate  attraction  of  all  of  them  will  be  the  same  as  if  all  were  Attraction  of 
compressed  at  the  center;  but  this  will  be  true  of  no  other  a  sphere. 
body.     The  earth  is  not  a  perfect  sphere,  and  two  lines  of 
attraction  from   distant  points  on  its  surface  may  not,  yea, 
will  not,  cross  each  other  at  the  earth's  center  of  gravity. 
(  See  Fig.  10.) 

(153.)  A  particle  anywhere  inside  of  a  spherical  shell  of     Attraction 
equal  thickness  and  density,  is  attracted  every  way  alike,  and  »n»ide  of 
of  course  would  show  no  indication  of  being  attracted  at  all. 
Hence  a  body  below  the  surface  of  the  earth,  as  in  a  deep  pit 
or  well,  will  be  less  attracted  than  on  the  surface,  as  it  will 
be  attracted  only  by  the  diminished  sphere  below  it.     At  the 
center  of  the  earth  a  body  would  be  attracted  by  the  earth 
every  way  alike,  ,and  there  would  be  no  unbalanced  force,  a  sphere. 
and  of  course  no  perceptible  or  sensible  attraction.* 

(  154  .)  The  attractive  power  on  the  surface  of  any  perfect    Ex  regsio 
and  homogeneous  sphere  may  be  expressed  by  the  mass  of  the  for  the   at- 
*phere  divided  by  the  square  of  the  radius.  tbe^TiLeo" 

Consider    the     earth  a  sphere  (as  it  is  very  nearly),  and  a  sphere. 
put  E  to  represent  its  mass,  and  r  its  mean  radius,  then 


E 

This  attractive  force,  algebraically  expressed  by  —  we  call  ^, 

and  it  is  sufficient  to  cause  bodies  to  fall  16T^  feet  during 
the  first  second  of  time.  If  the  earth  had  contained  more 
matter,  bodies  would  have  fallen  more  than  16T^  feet  the 
first  second  ;  if  less,  a  less  distance. 

With  the  same  matter,  but  more  compact,  so  that  r2  would  The  definit- 

...    ,  ..,     T-,  ,,  E  attraction  of 

be  less  with  E  the  same,  —  would  be  greater,  and  the  attrac-  ,he  earth. 

tive  power  at  the  surface  greater,  and  bodies  would  then  fall 
more  than  16-jL  feet  the  first  second  of  their  fall. 

Now  we  say  this  16T^  feet  is  the  measure  of  the  earth's 
attraction  at  its  surface,  and  it  is  made  the  unit  and  standard 
measure,  directly  or  indirectly,  for  all  astronomical  forces. 


*  See  Robinson's  Natural  Philosophy,  page  16 


172  ASTRONOMY. 

For  this  reason,  we  call  the  undivided  attention  to  this 
force,  the  known  —  the  noted  —  the  all-important 


TO  find  the  (  155.  )  By  the  theory  of  gravity,  we  can  readily  obtain  an 
*  °t  analytical  expression  for  the  attraction  of  a  sphere  at  any  dis- 

uy  distance,  tance  from  the  center,  after  knowing  the  attraction  at  the 
surface.  For  example.  Find  the  value  of  the  attraction  of 
the  earth,  at  the  distance  of  D  from  its  center  ;  r  being  the 
radius  of  the  earth,  and  g  the  gravity  at  the  surface  ;  put  x 
to  represent  the  attraction  sought.  Then  by  the  theory, 

9  •*••••    -'  0r'* 


As  g  and  r  are  constant  quantities,  the  variations  to  x  will 

correspond  entirely  to  the  variations  of  D2.     We  shall  often 

refer  to  this  equation. 

(156.)  As  every  particle  of  matter  in  the  universe  at- 
skm  for  the  tracts  every  other  particle,  therefore  the  moon  attracts  the 
"action  &of  earth  as  we^  as  *be  eartn  attracts  the  moon  ;  and  the  extent 
»wo  bodies,  by  which  they  will  draw  together,  depends  on  their  mutual  at- 

traction.    If  m  represents  the  mass  of  the  moon,  and  fi  the 

radius  of  the  lunar  orbit  ;  then, 

E 

The  earth  will  attract  the  moon  by  the  force  -^. 

The  moon  will  attract  the  earth  by  the  force  -^-. 

jfi  \  Im 
The  two  bodies  will  draw  together  by  the  force  —  ~  . 

If  we  substitute  the  value  of  g,  as  found  in  (  154  ),  in  equa- 

E 

tion  (5  ),  and  making  r  =  D,  then  we  have  the  expression  —  • 

The  spirit  of  these  expressions  will  be  more  apparent  when 
we  make  some  practical  applications  of  them,  as  we  intend 
soon  to  do. 


KEPLER'S    LAWS.  173 


CHAPTER    II. 

SEPLISB'S  LAWS  —  DEMONSTRATION  OF  THE  SECOND  AND  THIRD — 
HOW  A  PLANETARY  BODY  WILL  FIND  ITS  ORBIT. 

( 157. )  IN  tbis  chapter  we  design  to  make  some  examina-  CHAP.  n. 
tion  of  Kepler's  laws,  recapitulating  them  in  order. 

The  orbits   of  the  planets   are  ellipses,  having   the   sun   at 

,  J7    .       -     .  ler'a  laws 

wie  oj  weir  joci. 

This  law  is  but  a  concise  statement  of  an  observed  fact, 
which  never  could  have  been  drawn  from  any  other  source 
than  observation ;  but  the  second  law,  namely, 

That  the  radius  vector  of  any  planet  (  conceived  to  be  in  mo- 
tion )  sweeps  over  equal  areas  in  equal  times  is  susceptible  of 
a  rigid  mathematical  demonstration,  under  the  following  gen- 
eral theorem. 

Any  body,  being  in  motion,  and  constantly  urged  toward  any     A  gener»l 
fixed  point,  not  in  a  line  with  its  motion,  must  describe  equal 
areas  in  equal  times  round  that  point. 

Let  a  moving 
body  be  at  A, 
having  a  veloci- 
ty which  would  | 
carry  it  to  JB, 
say  in  one  sec- 
ond of  time.  By\ 
the  law  of  iner- 
ti.a,  it  would 
move  from  B  to 
C,  an  equal  dis- 1 

tance,  in  the  next  second  of  time.  But  during  this  second 
interval  of  time,  let  us  suppose  it  must  obey  an  impulse  or 
force  from  the  point  S,  sufficient  to  carry  it  to  D.  It  must 
then,  by  the  composition  of  forces  explained  in  natural  phi- 
losophy, describe  the  diagonal  B  E,  of  the  parallelogram 
BDEG. 


174  ASTRONOMY. 

CHAP  ii.  Now  in  the  first  interval  of  time,  we  supposed  the  moving 
body  described  the  triangle  SAB.  The  second  interval,  it 
would  have  described  the  triangle  S  £  C,  if  undisturbed  by 
any  force  at  S,  but  by  such  a  force  it  describes  the  triangle 
S  B  E\  but  the  triangle  S  B  E  is  equal  to  the  triangle 
SBC,  because  they  have  the  same  base  S  B,  and  lie  between 
the  parallels  S  B  and  E  C.  Also  the  triangle  S  B  C  is 
equal  to  the  triangle  S  A  B,  because  they  terminate  in  the 
same  point  S,  and  have  equal  bases  A  B  and  B  C.  There- 
fore the  triangle  S  A  B  is  equal  to  the  triangle  S  B  E,  be- 
cause they  are  both  equal  to  the  triangle  S  B  (7;  that  is,  the 
moving  body  describes  equal  areas  in  equal  times  about  the 
point  S,  and  this  is  entirely  independent  of  the  nature  of  the 
force  at  S\  it  may  be  directly  or  inversely  as  the  distance,  or 
as  the  square  of  the  distance. 

The    con-      The  converse  of  this  theorem  is,  that  when  a  body  describes 
r  the  equal  areas  in  equal  times  round  any  point,  the  body  is  con- 
stantly urged  toward  that  point;  and  therefore  as  the  planets 
are  observed  to  describe  equal  areas  in  equal  times  round  the 
sun,  their  tendency  is  toward  the  sun,  and  not  toward  any 
other  point  within  the  orbits. 
Kepler's      (158.)  The  third  law  of  Kepler  is  most  important  of  all, 

third      law  namely  —  The  squares  of  the  times  of  revolution  are  to  each 

prove*     that 

the  sun's  at-  °^er  as  ^e  cuoe&  °f  the  distances  from  the  sun.  By  this  law 
traction  is  it  is  proved,  that  it  is  the  same  force  which  urges  all  the 
theTuare  of  P^anets  to  tne  same  point,  and  that  its  intensity  is  inversely  as 
the  distance,  the  square  of  the  distance  from  that  point  (  the  center  of  the 
sun  ),  confirming  the  Newtonian  theory  of  gravity. 

Fig.  30.  To  show  this,  let  us  suppose  that  the 

planets  revolve  round  the  sun  in  circular 
orbits  ( which  is  not  far  from  the  truth), 
and  let  P  (  Fig.  30  )  represent  the  posi- 
tion of  a  planet ;  F  the  distance  which 
the  planet  is  drawn  from  a  tangent  during 
unity  of  time;  in  the  same  time  that  it 
describes  the  indefinite  small  arc  c;  and 
the  number  of  times  that  c  is  contained  in  the  whole  circum- 
ference, so  many  units  of  time,  then,  must  be  in  one  revolution. 


SOLAR   SYSTEM.  175 

If  D  is  the  diameter  of  the  orbit  and  fthe  time  of  revolu-    CHAf-  "• 
tion,  then  will 

"T  '  '  :  m 

So  for  any  other  planet.     If  f  is  the  force  urging  it  toward    AH  impor 
the  sun,  a  its  corresponding:  arc,  Tits  time  of  revolution,  and 
R  the  radius  of  its  orbit  ;  then,  reasoning  as  before, 


moustraUd. 


By  comparing  (  1  )  and  (  2  )  we  have 


I)2 
By  squaring,        f*  :  T2  :  :  —  : 

By  Kepler's  law,  t2  :  T2  :  :  r*  j  ^3. 
By  comparing  the  two  last  proportions,  and  observing  that 
2r  may  be  put  for  />,  and  reducing,  we  have 


But  by  the  well-known  property  of  the  circle,  we  have 

F  .  :  c  ::  c  :  2r;  or,   c2  =2rF. 
In  like  manner,         .         .         .  a2  =  2  Rf. 

Substituting  these  values  in  the  last  proportion,  and  redu 
cing,  we  have 


Or,         .         . 

Rf  :  rF  :: 

r  :  R. 

Hence, 

R2f=r2F', 

or,  F  if 

n  R2  :  r2  . 

1         i 

Or,         .        . 

• 

•        F'-f 

JL               A 
T"          J.C 

That  is ,  the  attractive  force  of  the  sun  is  reciprocally  pro- 
portional to  the  square  of  the  distance. 

(  159.)  If  we  commence  with  the  hypothesis,  that  bodies 
tend  toward  a  central  point  with  a  force  inversely  propor- 


176  ASTRONOMY. 

CHIP-  n-   tional  to  the  squares  of  their  distances,  and  then  compute 
and  laws  of  the  corresponding  times  of  revolution,  we  shall  find  that  the 
in    Kepler's  S9uares  °f  &•*  times  must  be  as  the  cubes  of  the  distances.  Hence 
third  law.      Kepler's  third  law  is  but  the  natural  mathematical  relation 
which  must  exist  between  times  and  distances  among  bodies 
moving  freely,  in   circular  orbits,  animated  by  one  central 
force  which  varies  as  the  inverse  square  of  the  distance. 
An  inquiry.      (  160. )  Having  shown  that  Kepler's   third  law  is  but  a 
mathematical  theorem  when  the  planets  move  in  circles  and 
their  masses  inappreciable  in  comparison  to  that  of  the  sun's, 
we  now  inquire  whether  the  law  is  true,  or  only  approximately 
true,  when  the  orbits  are  ellipses,   and  their  masses  consid- 
erable. 

HOW  answer-      On  one  of  these  points  of  inquiry,  the  reader  must  take  our 
ed-  assertion ;  for  its  demonstration  requires  the  use  of  the  inte- 

gral calculus,  a  method  that  we  designed  not  to  employ  in  thia 
work.     Kepler's  third  law  supposes  all  the  force  to  be  in  the 
central  body,  and  the  planets  only  moving  points.     But  wo 
have  seen  in  Art.  ( 156 )  that   the  attracting  force  on  any 
planet  is  the  mass  of  both  sun  and  planet  divided  by  the 
square   of  their    mutual  distance;  and  therefore   when  the 
mass  of  the  planet  is  appreciable,  the  force  is  increased,  and 
Masses  of  the  time  of  revolution  a  little  shortened.     But  the  fact  that 
the    planets  Kepler's  law  corresponds   so   well  with    other   observations 
compareT  to  proves     that  the  masses  of  all  the  planets  are  inappreciable 
the  sun.         compared  to  the  mass  of  the  sun. 

Kepler's      (  161. )  As  to  the  other  point,  we  state  distinctly  that  the 

third h.w ma-  planets  (considered  as  bodies  without  masses)  revolving  in 

^*[!y  ellipses  of  ever  so  great  eccentricity,  the  squares  of  the  times 

tic  orbits.      of  revolution  are  to  each  other  as  the  cubes  of  half  the  greater 

axes  of  the  orbits. 

We  shall  not  attempt  a  demonstration  of  this  truth ;  but 
hope,  the  following  explanation  will  give  the  reader  a  clear 
view  of  the  subject. 

Bodies  revolving  in  ellipses  round  one  of  the  foci,  may  be 
considered  to  have  a  rising  and  a  falling  motion;  something 
like  the  motion  of  a  pendulum.  The  motion  of  a  pendulum 
depends  on  the  force  of  gravity,  the  length  of  the  pendulum, 


PLANETARY  MOTION  177 

and  the  distance  the  pendulum  was  first  drawn  aside.     The    CHAP,  n, 

motion  of  a  planet  depends  on  the  force  of  gravity,  its  mean 

distance  from  the  sun,  and  the  original  impulse  firct  given  to    A  common 

it.     Most  persons,  who  have  not  investigated  this  subject,  wnwofopm- 

imagine  that  each  planet  must  originally  have  had  precisely 

the  impulse  it  did  have  to  maintain  itself  in  its  orbit ;  and  so 

it  must,  to  maintain  itself  in  just  that  definite  orbit  in  which 

it  moves.     But  had  the  original  impulse  been  different,  either  as 

to  amount  or  direction,  or  as  to  both,  then  lytlie  action  of  gravity 

and  inertia,  the  planet  would  have  found  a  corresponding  orbit. 

(162.)  The  force  of  gravity,  from  the  action  of  any  attract-  Examina. 
ing  body,  is  always  as  the  mass  of  the  body  divided  by  the  square  tion  of  th* 
of  its  distance.  Algebraically,  if  Mia  the  masa  of  the  body,  ^tumi^  » 
r  its  distance,  and  F  the  force  at  that  distance,  then  (see  156)  «"'ptte  o*J«« 

we  have  -  ,   -       -  ^-=F.         (See  Fig.  28.) 

Now  if  the  planet  has  such  a  velocity,  c,  as  to  correspond 
with  the  proportion  F    :     c     :  :     c    :     2r, 

Or,-       -       -       -     c=J2rF=^ ,  and  that  velocity  at 

right  angles  to  r  (Fig.  31),  then  the  planet's  orbit  would  be  a 
circle,  with  the  radius  r.     If  the  velocity  had  been  less  in 
amount  than  this  expression,  and  still  at  right  angles  to  r,  then 
the  planet  would  fall  within  the  circle,  and  the  action  of  gra- 
vity would  increase  the  motion  of  the  planet ;  and  the  motion 
would  increase  faster  than  the  increased  action  of  gravity : 
there  would  be  a  point,  then,  where  the  motion  would  be  suflicient 
to  maintain  the  planet  in  a  circle,  at  its  then  distance  ;  but  the  **>•  ™n.  B«. 
direction  of  the  motion  will  not  permit  the  planet  to  run  into  ™*  £  ****" 
the  circle,  and  it  must  fall  within  it. 

The  motion  continues  to  increase  until  its  position  becomes 
at  right  angles  to  the  radius  vector ;  the  motion  is  then  as 
much  more  than  sufficient  to  maintain  the  planet  in  a  circle, 
as  it  was  insufficient  in  the  first  instance;  it  therefore  rises, 
by  the  law  of  inertia,  and  returns  to  the  original  point  P, 
where  it  will  have  the  same  velocity  as  before ;  and  thus  the 
planet  vibrates  between  two  extreme  distances. 
12 


«7S 


ASTRONOMY. 


mean  distan- 
ces of  the  or 

\»its. 


A  hypothe 
deal  case. 


CHAP,  ii  If  the  velocity,  on  starting  from  the  point  P,  were  very 
Gravity  and  much  less  than  sufficient  to  maintain  a  circle,  at  that  distance, 
original  ve  ^hen  ^he  orbit  it  would  take  would  be  very  eccentric,  and 

locity    deter-  . 

mine  the  ec-  its  mean  distance  much  less  than  r.     If  the  original  velocity 
at  P  were  greater  than  to  maintain  it  in  a  circle,  it  would 
"  pass  outside  of  this  circle,  and  the  point  P  would  be  the  peri- 
helion point  of  the  orbit. 

Thus,  we  perceive,  that  the  eccentricity  of  orbits  and  mean 
distances  from  the  sun  depend  on  the  amount  and  direction 
of  the  original  impulse,  or  velocity  which  the  planet  has  in 
some  way  obtained;  and  it  is  not  necessary  that  the  planet 
should  have  any  definite  impulse,  either  in  amount  or  direction,  to 
move  in  an  orbit,  if  the  direction  is  not  directly  to  or  from  the  sun 
(163.)  For  a  more  definite  explanation  of  this  subject,  let 
us  conceive  a  planet  launched  out  into  space  with  a  velocity 
sufficient  to  maintain  it  in  a  circle  at  the  distance  it  then  hap- 
pened to  be,  but  the  direction  of  such  velocity  not  at  right 
angles  to  the  sun,  then  the  orbit  will  be  elliptical,  and  the 
degree  of  eccentricity  will  depend  on  the  direction  of  the 
motion ;  but  the  longer  axis  of  the  orbit  will  be  equal  to  the 
diameter  of  the  circle,  to  which  its  velocity  corresponds ;  and 

the  time  of  its  revolution  will  be 
the  same,  whether  the  orbit  is 
circular  or  more  or  less  elliptical. 
Let  P  (Fig.  31)  be  the  posi- 
tion of  a  planet,  S  the  sun;  and 
let  the  velocity,  a,  be  just  suffi- 
cient to  maintain  the  planet  in 
a  circle,  if  it  were  at  right  angles 
toSP. 

Now  to  find  the  orbit  that  this 
planet  would  describe,  draw  the 
line  P  0  at  right  angles  to  a, 
and  from  S  let  fall  a  perpendi- 
cular on  PC;  SC  will  be  the 
eccentricity  of  the  orbit,  and  PC 
will  be  the  half  of  its  conjugate 
axis;  and  with  these  lines  the  whole  orbit  is  known. 


PLANETARY    MOTION. 


170 


(164.)  Now  let  us  suppose  that  a  planet  is  rather  carelessly    CHAP,  n 
launched  into  space,  with  a  velocity  neither  at  right  angles  to         pianeti 
the  sun,  nor  of  sufficient  amount  to  maintain  it  in  a  circle,  at  w«llfindtheu 

orbits,  what- 

that  distance  from  the  sun.  ever  be  the 

Let  P  (Fig.  32)  represent  the  Fig.  32.  direction  and 

\-     .1,  1  Ai      •••••^^••MBBI^     forceo'  theii 

position    of    the    planet,  a    thei 
amount  and  direction  of  its  hap- 
hazard  velocity   during  the  first  I 
unit  of  time.     The  direction  of 
the  motion  being  within  a  right  I 
angle  to  S  P,  the  action  of  gra- 
vity increases 
the     velocity  TJ/--N 
of  the  planet,      >^ 
on  the  same 

principle  that  a  falling  body  in- 
creases in  velocity ;  and  the  planet  I 
goes  on  in  a  curve, describing  equal 
areas  in  equal  times  round  the  point 
S;  and  it  will  find  a  point,  p,  where 
its  increased  velocity  will  be  justj 

equal  to  the  velocity  in  a  circle  whose  radius  is  the  diminished 
distance  Sp.  From  the  point  p,  and  at  right  angles  to  «, 
draw  p  C,  £c.,  forming  the  right  angled  triangle  p  C  S.  SO 
is  the  eccentricity,  Sa  the  mean  distance,  and  p  C  half  the 
conjugate  axis  of  the  orbit. 

If  the  planet  is  launched  into  space  in  the  other  direction,     The 
the  action  of  gravity  will  diminish  its  motion,  and  will  bring  wilt  be  sym" 

•    r  .  i  i  metrical     on 

it  at  right  angles  to  the  line  joining  the  sun ;  it  is  then  at  its  each  side  of 
apogee,  with  a  motion  too  feeble  to  maintain  a  circle  at  that  aP°see   anfl 
distance ;  and  it  will,  of  course,  approach  nearer  and  nearer  pe 
to  the  sun  by  the  same  laws  of  motion  and  force  that  it  receded 
from  the  sun ;  hence  the  curve  on  each  side  of  the  apogee 
will  be  symmetrical ;  and  the  same  reasoning  will  apply  to  the 
curve  on  each  side  of  the  perigee ;  and,  in  short,  we  shall 
have  an  ellipse. 

To  sum  up  the  whole  matter,  it  is  found  by  a  strict  exanii-    AB 
nation  of  the  laws  <&  gravity,  motion,  and  inertia,  that  whatever  ,*on. 


180  ASTRONOMY. 

CHAP.  II.  may  be  the  primary  force  and  direction  given  to  a  planetary 
body  ( if  not  directly  to  or  from  the  sun  ),  the  planet  mil  find 
a  corresponding  orbit,  of  a  greater  or  less  eccentricity,  and  of  a 
greater  or  less  mean  distance ;  and  whatever  be  the  eccentricity 
of  the.  orbit,  the  real  velocity,  at  tJie  extremity  of  the  shorter  axis, 
will  be  just  sufficient  to  maintain  the  planet  in  a  circular  orbit,  at 
that  mean  distance  from  the  sun* 

Theory  of       *  Let  S  be  the  sun,  and  P  the  position  of  a  planet  as  repre- 

l)r.      Gibers  .    .  * 

concerning     sentecl  m  ™e  annexed  ngure,  and  we  may  now  suppose  it  to 
the  asteroid*  burst  into  fragments,  the  figure  representing  three  fragments 

only;  the  velocity  and  direction  of  one  represented  by  a;  of 

another  by  b,  and  of  a  third  by  c,  &c. 
Flff.  33. 


As  action  is  just  equal  to  reaction,  under  all  circumstances, 
therefore  the  bursting  of  a  planet  can  give  the  whole  mass  no 
additional  velocity  ;  a  small  mass  may  be  blown  off  at  a  great 
velocity,  but  there  will  be  an  equal  reaction  on  other  masses, 


On  th«  direction. 


bursting  of  a 

planet,    the      The  whole  might  simply  burst  into  about  equal  parts,  and 
fragments      then  they  would  but  separate,  and  all  the  parts  move  along 

would     take  ,          J  i.ii 

orbits  corre-  m  *"e  same  general  direction,  and  with  the  same  aggregate 
spending  to  velocity  as  the  original  planet.     The  bursting  of  a  rocket  is 

a  ™ry  minute,  but  a  very  faithful  representation  of  such  an 

explosion 


KEPLER'S    LAWS.  181 

(  165.)  To  see  whether  Kepler's  third  law  applies  to  ellipses,  CHAP.  n. 
we  represent  half  the  greater  axis  of  any  ellipse  by  -4,  and  Kepler^ 
half  the  shorter  axis  by  B,  and  then  (3.1416)^4^  is  the  area  third  law  ri- 
of  the  ellipse.  Also,  let  a  represent  the  velocity  or  distance  f°  ^J^™ 

~~   ellipses,      as 

If  the  velocities  of  the  several  fragments  were  equal,  the  well  a«  to 
times  of  their  revolutions  would  be  equal ;  but  the  eccentri-  circle»- 
cities  of  the  several  orbits  would  depend  on  the  angles  of  a, 
6,  c,  &c.,  with  S  P.  If  a  is  at  right  angles  to  <S  P,  and  just 
sufficient  to  maintain  the  planet  in  a  circle  at  that  distance, 
then  its  orbit  would  have  no  eccentricity.  If  still  at  right 
angles,  but  not  sufficient  to  maintain  a  circle  at  that  distance, 
then  SP  would  be  the  greatest  radius  of  the  orbit.  Hence, 
we  perceive,  there  is  an  abundance  of  room  to  have  a  multi- 
tude of  orbits  passing  through  the  same  point,  during  the 
first  one*  or  two  revolutions ;  and  the  times  of  such  revolu- 
tions may  be  equal,  or  very  unequal.  In  short,  there  is  no 
physical  impossibility  to  be  urged  against  the  theory  of  Dr. 
Olbers,  that  the  asteroids  are  but  fragments  of  a  planet. 

The  objection  is  ( if  an  objection  it  can  be  called )  i  hat 
these  planets  have  not,  in  fact,  a  common  node,  nor  have  an 
approximation  to  one ;  nor  have  they  an  approximation  1 3  a 
common  radius  vector,  as  S  P.  But  the  objection  vanishes 
when  we  consider  that  the  elements  of  the  different  orbits 
must  be  variable ;  and  time,  a  sufficient  length  of  time,  would 
separate  the  nodes  and  change  the  positions  of  the  orbits  so 
as  to  hide  the  common  origin,  as  is  now  the  case. 

But  if  it  be  true  that  these  planets  once  had  a  common 
origin  in  one  large  planet,  it  is  possible  to  find  the  variable 
nature  of  the  elements  of  their  orbits  to  such  a  degree  of 
exactness  as  to  trace  them  back  to  that  origin  —  define  the 
place  where,  and  the  time  when,  the  separation  must  have 
occurred. 

If,  however,  a  planet  should  burst  at  one  time,  and  after- 
ward one  or  more  of  the  fragments  burst,  there  could  be  no 
tracing  to  a  common  origin ;  hence  it  is  possible  that  the 
asteroids  in  question  may  have  a  common  origin,  and  it  be 
wholly  beyond  the  power  of  man  to  show  it. 


182  ASTRONOMY. 

CHAP.  ii.  that  the  planet  will  move  in  a  unit  of  time,  when  at  the  ex- 
tremity of  its  shorter  axis;  then  ±aB  will  express  the  area 
described  in  that  unit  of  time. 

But  as  equal  areas  are  described  in  equal  times,  as  often 
as  this  area  is  contained  in  the  whole  ellipse  will  be  the  num- 
ber of  such  units  in  a  revolution.  Put  /=  that  number,  or 
the  time  of  revolution ;  then 

(3.1416)  Aff      2(3.1416)^4 

JL  ft  J)  ft 

Let  A'  and  B'  be  the  semiaxes  of  any  other  ellipse ;  a'  the 
velocity  at  the  extremity  of  B',  and  t'  the  time  of  revolution ; 

then  will  t  = : • 

a 

By  comparing  these  equations,  and  rejecting  common  fac- 

A          A' 

tors,  we  have      -     t    :     t     :  :     —     :     —7. 

a  a 


I.)  Ttf  I'^Af 

But  by  Art.  162,  a=J-T,    and    af=J^77 

"   A  "  -A 

M  mass  of  sun) ;  and  putting  the  values  of  a  and  a',  in  the 
above  proportion,  we  have 

<     :     ,     !S     AJj^      :     £^'; 

Or,  -       -       /    :    f     ::      AJA        :    A'JA'. 

By  squaring  t2  :    t'*  ::         A3          :      A'3-,     which    is 
Kepler's  third  law. 

Eccentrici.      (166.)  We  have  seen,  in  articles  163  and  164,  that  the 
f  the  eccentricity  of  an  orbit  depends  on  the  direction  of  the  motion 
bits  change  to  the  radius  vector,  when  the  planet  is  at  mean  distance.     If 
by  their  mu.  that  direction  is  at  right  angles  to  the  radius  vector  at  that 
101  time,  then  the  eccentricity  is  nothing.     If  its  direction  is  very 
acute,  then  the  eccentricity  is  very  great,  &c. 

Now  suppose  another  planet  to  be  situated  at  B  (Fig.  32); 
its  attraction  on  the  planet,  passing  along  in  the  orbit  p  a,  ii 
to  give  the  velocity,  a,  a  direction  more  at  right  angles  to 


KEPLER'S  LAWS. 


183 


Sp,  and  thus  to  diminish  the  eccentricity  of  the  orbit.  If 
the  disturbing  body,  B,  were  anywhere  near  the  line  C  S,  its 
tendency  would  be  to  increase  the  eccentricity;  and  thus,  in 
general,  A  disturbing  body  near  a  line  of  the  shorter  axis  of 
an  orbit,  has  a  tendency  to  diminish  the  eccentricity  of  the  orbit 
of  the  disturbed  body  ;  and,  anywhere  near  a  line  of  the  greater 
axis,  has  a  tendency  to  increase  the  eccentricity.  Hence  the 
eccentricities  of  the  planets  change  in  consequence  of  their 
mutual  attractions;  but  their  mean  distances  never  change. 

(167.)  As  the  time  of  revolution  is  always  the  same  for 
the  same  mean  distance,  whatever  be  the  eccentricity  of  the 
orbit,  therefore  if  we  conceive  a  planet  to  turn  into  an  infi- 
nitely eccentric  orbit,  and  fall  directly  to  the  sun,  the  time  of 
such  fall  would  be  half  a  revolution,  in  an  orbit  of  half  its 
present  mean  distance,  as  we  perceive,  by  inspecting  Fig.  34. 

Hence,  by  Kepler's  third  law,  we  can  compute  the  „. 
time  that  would  be  required  for  any  planet  to  fall  to 
the  sun.     Let  x  represent  the  time  a  planet  would 
revolve  in  this  new  and  infinitely  eccentric  orbit ;  then, 
by  Kepler's  law, 

t*  :  x*  ::  23  :  1»,  or,     **=%-. 

o 

Therefore  half  of  the  revolution,  or  simply  the  time 
of  the  fall,  must  be  expressed  by   - — - ,     or, 


CHAP.  11. 

The  mean 
distanoe»  »•• 


x-x 

/      \ 


The    prin 

ciples  and 
the  computa. 
tion  of  the 
time  required 
for  the  plan- 
ets to  fall  t« 
the  sim. 


that  is,  to  find  the  time  in  which  any  planet  would 
fall  to  the  sun,  if  simply  abandoned  to  its  gravity,  or  the  time 
in  which  any  secondary  planet  would  fall  to  its  primary,  divide 
its  time  of  revolution  by  four  times  the  square  root  of  two. 
By  applying  this  rule,  we  find  that 

Mercury  would  fall  to  the  sun  in 

Venus, 

Earth, 

Mars 

Jupiter, 

Saturn 1901 

Uranus, 5424 

The  moon  would  fall  to  the  earth  in  4d.  19  h.  54m.  36  •. 


Days.  h. 
15  13 

ra. 
13 

39  17 

19 

64  13 

39 

121  10 

36 

765  21 

36 

1901  23 

24 

5424  16 

52 

184  ASTRONOMY. 


CHAPTER  III. 

MASSES    OF    THE    PLANETS DENSITIES PRESSURE    ON     THHK 

SURFACES. 

CHAP.  in.       ( 1^8. )    IF  the   earth  contained  more   matter,  it  would 

Masses  me  a  afctracfc  ^th   grater  force ;  and  if  the    sun   has  a   greater 

•nred  by  at-  power  of  attraction  than  the  earth,  it  is  because  it  contains 

traction.       more  matter  than  the  earth ;  and  therefore,  if  we  can  find  the 

relative   degree  of  attraction  between  two  bodies,  we  have 

their  relative  masses  of  matter. 

If  the  earth  and  sun  have  the  same  amount  of  matter,  they 
will  attract  equally  at  equal  distances.  Let  M\>e  the  mass 
of  the  sun,  and  E  the  mass  of  the  earth,  then  (  at  the  same 
unit  of  distance),  the  attraction  of  the  sun  is,  to  tlie  attraction  of 
the  earth,  as  M  to  E. 

But  attraction  is  inversely  as  the  square  of  the  distance. 

M 
Hence  the  attraction  of  the  sun  at  D  distance,  is  -=—  ;    and 

E 
the  attraction  of  the  earth  at  R  distance  is  ~Ra* 

Gravity  of  The  earth  is  made  to  deviate  from  a  tangent  of  its  orbit 
the  snn  is  ^j  the  attraction  of  the  sun;  and  the  moon  is  made  to  deviate 
the  devia.  from  a  tangent  of  its  orbit  by  the  attraction  of  the  earth,  and 
tion  of  the  the  amount  of  these  deviations  will  give  the  respective 
tangent  of  iu  amounts  of  solar  and  terrestrial  gravity, 
orbit.  If  we  take  any  small  period  of  time,  as  a  minute  or  a  sec- 

ond, and  compute  the  versed  sine  of  the  arc  which  the  earth 
describes  in  its  orbit  during  that  time,  such  a  quantity  will 
express  the  sun's  attraction ;  and  if  we  compute  the  versed 
sine  of  the  arc  which   the  moon  describes  in  the  same  time, 
that  quantity  will  express  the  attraction  of  the  earth. 
HOW  to  com-      in  Figure  30,  Art.  158,  ^represents  the  versed  sine  of  an 
parative°0in  arc ;  and  if  we  take  D  to  represent  the  mean  distance  be- 
masses  of  the  twcen  the   earth  and  sun,  and  consider  the  orbit    a  circle 
*tm  and  earth  ^ag  we  may  wjthout  error,  164),  the  whole  circumference  is 


MASSES  OF  THE  PLANETS.  185 

it]}  (*•  =  6.2832).     Divide  the  whole  circumference  by  the  CHAP.  m. 
number  of  minutes  in  a  revolution ;   say  T,  and  the  quotient 
will  represent  the  arc  a  (Fig.  30).     When  T  is  very  large, 
and  of  course  a  very  small,  the  chord  and  arc  practically  coin- 
cide ;  and  by  the  well  known  property  of  the  circle,  we  have 


2D  :  a::  a:  F;     Or,     F=  ~.       .  (1) 

_  *D  **D2  a3          VK* 

*      But  a  =  -^-;  hence,  a*  =  -^-,     and  ^=  —  ; 

That   is,  F  =     _  a  ;  which  is  an  expression  for  the  sun's 

attraction  at  the  distance  of  the  earth.     But  -=r—  is  also  an 
expression   for  the  sun's  attraction  at  the  same  distance  ; 


therefore,  _  =  -^-;     Or,     #**- 


In  the  same  manner,  if  R  represents  the  radius  of  the  lunar 
orbit;  t  the  number  of  minutes  in  the  revolution  of  the 
moon  ;  the  mass  of  the  central  attracting  body  (  in  this  case 
the  earth  )  must  be  expressed  by 


Therefore,  E  :  M:  :    —  :  -^-. 

This  proportion  gives  a  relation  between  the  masses  of  the 
earth  and  sun  expressed  in  known  quantities. 

If  we  assume  unity  for  the  mass  of  the  earth,  we  shall 
have  for  the  mass  of  the  sun, 

.        .        .     (A) 

(169.)  This  is  a  very  general  equation,  for  D  may  repre-   The  general 
sent  the  radius  of  the  earth's  orbit,  or  the  orbit  of  Jupiter  or  application 
Saturn,  arid  3Twill  be  the  corresponding  time  of  revolution.  °ioB/* 
Also  R  may  represent  the  radius  of  the  lunar  orbit,  or  the 


186  ASTRONOMY. 

CHAP.  in.  orbit  of  one  of  Jupiter's  or  Saturn's  moons,  and  then  t  will 

be  its  corresponding  time  of  revolution. 

The  results      This  equation,  however,  is  not  one  of  strict  accuracy,  as 
of  the  eqna-  f^g  distance  a  planet  falls  from  the  tangent  of  its  orbit,  in  a 

tion  will  not 


accu™tend  definite  moment   of  time,  is  not,  accurately  j.  -,  but  —  ~— 

why  ?  -°a 

(  see  156  ),  E  being  the  mass  of  the  planet.  The  force 
which  retains  a  moon  in  its  orbit  is  not  only  the  attracting 
mass  of  the  central  body,  but  that  of  the  moon  also.  Bu? 
the  planets  being  very  small  in  relation  to  the  sun,  and  in 
general  the  masses  of  satellites  being  very  small  in  respect  to 
their  primaries,  the  errors  in  using  this  equation  will  in  gen- 
eral be  very  small.  The  error  will  be  greatest  in  obtaining 

Corrections 

for  equation  the  mass  ot  the  earth,  as  in  that  case  the  equation  involves 
<A)  the  periodic  time  of  the  moon;  which  period  is  different  from 

what  it  would  be  were  the  moon  governed  by  the  attraction 
of  the  earth  alone  ;  but  the  mass  of  the  moon  is  no  inconsid- 
erable part  of  the  entire  mass  of  both  earth  and  moon  ;  and 
also  the  attraction  of  the  sun  on  the  combined  mass  of  tho 
earth  and  moon,  prolongs  the  moon's  periodical  time  by  about 
its  179th  part. 

With  these  corrections  the  equation  will  give  the  mass  of 
the  sun  to  a  great  degree  of  accuracy  ;  but  we  can  determine 
the  mass  of  the  sun  by  the  following  method  : 

From  Art.  155,  we  learn  that  the  attraction  of  the  earth 


curate   «(«a-  /    3   \ 

Uom*  at   the  distance   to  the  sun,  is    g  \j^j- 


By  Art.  168,  we  have  just  seen  that  the  attraction  of  the 

fr2  D 

sun  on  the  earth,  is     =    ;  therefore, 


Taking  the  mass  of  the  earth  as  unity,  we  have 

W2  7)3 


Equation   (J?)   is  more   accurate  than  equation   (.4), 


MASSES  OF  THE  PLANETS.  187 

because  (  B  )  does  not  involve  the  periodical  revolution  of  the   CM  IP.  in. 
moon,  which  requires  correction  to  free  it  from  the  effects  of 
the  sun's  attraction.     To  obtain  a  numerical  expression  for  HOW  to  ob. 
the  mass  of  the  sun,  M,  the  numerator  and  denominator  of  the  ™       " 
right  hand  member  of  equation  (  B  )  must  be  rendered  homo-  »nit. 
geneous  ;   and  as  g,  the  force  of  gravity  of  the  earth,  is  ex- 
pressed in  feet  (  corresponding  to  T  in  seconds  ),  therefore  r 
the  mean  radius  of  the  earth,  and  D  the  distance  to  the  sun, 
must  be  expressed  in  feet.     But  from  the  sun's  horizontal 
parallax,  we  have  the  ratio  between  r  and  D  (  see  127  ), 
which  gives  D  =  23984  r. 

This  reduces  the  fraction  to  -  fT~rjia  -  •      But  *°  ex~ 

press  the  whole  in  numbers,  we  must  give  each  symbol  its 
value  ;  that  is,  *  =  6.2832  ;  r  =  (  3956  )  (  5280  )  ;  g  =  16.1  ; 
T=  31558150,  the  number  of  seconds  in  a  sidereal  year. 
(6.2832)2(23984W3956)(5280) 


It  would  be  too  tedious  to  carry  this  out,  arithmetically,  An 

without  the  aid  of  logarithms,  and  accordingly  we  give  the  showing 

i        -^      L-    i     i  *-       *v  sreat  ut 

loganthmetical  solution,  thus,  of  logarithms 

6  .2832  log.  0.798178X2      .        .  .      1.596356 

23984    log.  4.380000X3         .        .  13.140000 

3956   log.            .        .        .        .  .     3  .597256 

5280   log.        .....  3  .722632 

Logarithm  of  the  numerator,     .        .  .22  .056244 

32.2   log  ......  1  .507856          Themasaol 

31558150  log.  7.499114X2       .        .        .14  .998228     the  '«  d" 

-  terminal. 

Logarithm  of  the  denominator,       .  16  .506084 

Therefore  M=  354945,  whose  log.  is  5  .550160 

That  is,  the  mass  or  force  of  attraction  in  the  sun  is 
354945  times  the  mass  or  attraction  of  the  earth.    La  Place 


188  ASTRONOMY. 

CHAP,  m.  says  it  is  354936  times ;  but  the  difference  is  of  no  conse- 
quence. 

Equation  (A)  gives  350750;  but  equation  (-5),  as  we 
have  before  remarked,  is  far  more  accurate,  and  the  result 
here  given,  agrees,  within  a  few  units,  with  the  best  author- 
ities. 

Equation  ( B )  is  not  general ;  it  will  only  apply  to  the 
relative  masses  of  sun  and  moon,  because  we  do  not  know 
the  element  g,  the  attraction,  on  the  surface  of  any  other 
planet,  except  the  earth.  That  is,  we  do  not  know  it  as  a 
primary  fact ;  we  can  deduce  it  after  we  shall  have  determined 
the  mass  of  a  planet. 

Equation  (  A)  is  general,  and  although  not  accurate,  when 

applied  to  the  earth  and  sun,  is  sufficiently  so  when  applied 

to  finding  the  masses  of  Jupiter,  Saturn,  or  Uranus ;  because 

these  planets  are  so  remote  from  the  sun,  that  the  revolutions 

of  their  satellites  are  not  troubled  by  the  sun's  attraction. 

TO  find  the      (170.)    To  find  the  mass  of  Jupiter   (or  which  is  the 

masses  of  Ju-  game  ^jng  fae  mass  Of  fae  gun  w^en  JUpiter  is  taken  as 

piter,  Saturn, 

and  Uranus,  unity),  we  conceive  the  earth  to  be  a  moon  revolving  about  the 
sun,  and  compare  it  with  one  of  Jupiter's  satellites  revolving 
round  that  body.  To  apply  equation  (/I),  let  the  radius  of  the 
earth  equal  unify,  then  the  radius  of  Jupiter  must  be  11.11 
(Art.  131  ) ;  and  by  observation  the  orbit  radius  of  Jupiter's 
4th  satellite  is  26.9983  times  Jupiter's  radius,  therefore 
the  distance  from  the  center  of  Jupiter  to  the  orbit  of  its 
4th  satellite,  must  be  the  following  product  (11.11)  (26.9983), 
which  corresponds  to  R  in  the  equation.  D  =  23984; 
T=  365.256;  t=  16.6888. 

Therefore,  by  applying  equation  (A),  (M= 
(16.6888)2  (23984)3 


have  Jf= 


(365.256)2(1L11)3(26 .9983)'' 

By  logarithms  16.6888  log.  1  .222410x2  .    2  .444820 
23984  log.  4  .380000x3  . 13  .140000 

Logarithm  of  the  numerator,  .         15.  584820 


MASSES   OF   THE   PLANETS.  18* 

865.256,  log.  2  .562600x2    .  5  .125200       CHAP.  ID 

11.11,  log.  1.045714x3    .  3.137142 
26.9983,  log.  1.431320x3    .  4  .293960 

Logarithm  of  the  denominator,        .         .       12  .556302 

Therefore  M  =  1068*,  log.    ...         3  .028518 

This  result  shows  that  the  mass  of  the  sun  is  1068  times 
the  mass  of  Jupiter ;  but  we  previously  found  the  mass  of 
the  sun  to  be  354945  times  the  mass  of  the  earth,  and  if 
unity  is  taken  for  the  mass  of  the  earth,  and  J  for  the  mass 
of  Jupiter,  we  shall  have 

1068  J=  354945; 

because  each  member  of  this  equation  is  equal  to  the  mass 
of  the  sun. 

By  dividing  both  members  of  this  equation  by  1068,  we  The  mass  of 
find  J,he  mass  of  Jupiter  to  be  332  times  that  of  the  earth ;   "red",,0^ 
but'in  Art.  132,  we  found  the  bulk  of  Jupiter  to  be  1260  of  the  earth, 
times  the  bulk  of  the  earth ;  therefore  the  density  of  Jupiter 
is  much  less  than  the  density  of  the  earth. 

In  the  same  manner  we  may  find  the  masses  of  Saturn  and  The  masses 
Uranus  —  the  former  is  105.6  times,  and  the  latter  18.2  of  Satnra 
times  the  mass  of  the  earth.  and  Uranu*' 

The  principles  embraced  in  equation  ( A )  apply  only  to 
those  planets  that  have  satellites ;  for  it  is  by  the  rapid  or 
slow  motion  of  such  satellites  that  we  determine  the  amount 
of  the  attractive  force  of  the  planet. 

In  short,  the  masses  of  those  planets  which  have  satellites,  what  re- 
are  known  to  groat  accuracy;  but  the  results  attached  to  sults  may  *** 
others  in  table  IIT,  must  be  regarded  as  near  approximations.  accurate. 

The  slight  variations  which  the  earth's  motion  experiences    The  m 
by  the  attractions  of  Venus  and  Mars,  are  sufficiently  sensi- 
ble   to  make  known   the  masses   of  these  planets;   and   M.  Mercnrj 
Burckhardt  gives  ^^VTT  for  Venus,  and  ??T£_?T  for  Mars 
( the  mass  of  the  sun  being  unity ) .  Mercury  he  put  down  at 


•  This  is  a  correct  result  according  to  these  data;  but  more  modern 
observations,  in  relation  to  the  micrometic  measure  of  Jupiter,  and 
the  distance  of  his  satellites,  ^ive  results  a  little  different,  as  expressed 
in  table  III. 


190  ASTRONOMY. 

CHAP,  ui  _-r2i-T_;  but  this  result  is  little  more  than  hypothetical, 
as  it  is  drawn  from  its  volume,  on  the  supposition  that  the 
densities  of  the  planets  are  reciprocal  to  their  mean  distances 
from  the  sun;  which  is  nearly  true  for  Venus,  the  earth,  and 
Mars. 


of  (171.)  It  may  be  astonishing,  but  it  is  nevertheless  true, 
the  lunar  par.  that  by  means  of  equations  (-4)  and  (B~)  we  can  find  the 
alia*,  we  diameter  of  the  earth  to  a  greater  degree  of  exactness  than  by 

may  find  the  . 

diameter  of  anJ  one  ac^ual  measurement. 

the  earth.  "We  have  several  times  observed  that  equation  (  A  )  is  not 
accurate  when  used  to  find  the  masses  of  the  earth  and  sun, 
because  it  contained  the  time  of  the  revolution  of  the  moon; 
which  revolution  is  accelerated  by  the  gravity  of  the  moon,  and 
retarded  by  the  action  of  the  sun. 

Therefore,  to  make  equation  (  A  )  accurately  express  the 
mass  of  the  sun,  the  element  t2  requires  two  corrections, 
which  will  be  determined  by  subsequent  investigation.  The 
first  is  an  increase  of  Tl  jth  part  ;  the  second  is  a  diminution 
of  ^|jth  part,  and  both  corrections  will  be  made  if  we  take 

76-357 

^K  ogQ*2  m  place  of  t2. 

75-358 

A  common      Then  having  two  correct  expressions  for  the  mass  of  the 
sun,  those  two  expressions  must  equal  each  other  ;  that  is, 
76-357 

75-358 

By  suppressing  common  factors,  we  have 
76-357  12          ** 


75-358  R* 

In  this  equation  r  represents  the  mean  radius  of  the  earth, 
and  we  will  suppose  it  unknown  ;  the  equation  will  then 
make  it  known. 

The  relation  between  JB,  the  mean  radius  of  the  lunar  or- 
bit, and  r,  the  mean  radius  of  the  earth,  is  given  by  means 
of  the  moon's  horizontal  parallax. 

"^e  moon's  equatorial  horizontal  parallax,  as  we  nave  seen, 
and  (65)  is  57'  3";  but  the  horizontal  parallax  for  the  mean  ra- 


MASSES    OF    THE  PLANETS.  191 

dius,  is  56'  57";  this  makes  R  =  (  60.36  )  r,  whatever  the  CHAP.  m. 
numerical  value  of  r  may  be.     Put   this  value  of  R  in  the 


preceding   equation,  and   suppress   the   common    factor  r2,  zontai 

76-357  *2  *2 

we  then  have         ___=- 

Therefore,  ' 


75-358(60.36)3**  * 

As  g  is  expressed  in  feet,  and  corresponds  to  t  in  seconds,  confidence 
the  numerical  value  of  T  will  be  in  feet,  which  divided  by  in  the  result< 
5280,  the  number  of  feet  in  a  mile,  will  give  the  number  of 
miles  in  the  mean  radius  or  mean  semidiameter  of  the  earth ; 
and  by  applying  the  preceding  equation,  giving  g,  t,  and  »,  their 
proper  values ;  and  by  the  help  of  logarithms,  we  readily  find 
r  =  3955.8  miles  ;  less  than  a  mile  from  the  most  approved 
result ;  and  we  do  not  hesitate  to  say,  that  this  result  is  more 
to  be  relied  upon  than  any  other. 

MASS   OF   THE   MOON. 

( 172. )  Approximations  to  the  mass  of  the  moon  hav<*  The  ma3-  ^ 
been  determined,  from  time  to  time,  by  careful  observations  the      moon 
on  the  tides ;  but  it  is  in  vain  to  look  for  mathematical  re-  determined 
suits  from  this  source  ;  for  it  is  impossible  to  decide  whether  from    obser. 

vations      i 
the  tides. 


any  particular  tide  has  been  accelerated  or  retarded,  aug-  va 


mented  or  diminished,  by  the  winds  and  weather;  and  if  not 
affected  at  the  place  of  observation,  it  might  have  been  at 
remote  distances ;  but  notwithstanding  this  objection,  the 
mass  of  the  moon  can  be  pretty  accurately  determined  by 
means  of  the  tides,  owing  to  the  great  number  and  variety 
of  observations  that  can  be  brought  into  the  account;  and 
we  shall  give  an  exposition  of  this  deduction  hereafter ;  but 
at  present  we  shall  confine  our  attention  to  the  following 
simple  and  elegant  method  of  obtaining  the  same  result. 

If  the  moon  had  no  mass ;  that  is,  if  it  were  a  mere  mate- 
rial point,  and  was  not  disturbed  by  the  attraction  of  the 
sun,  then  the  distance  that  the  moon  would  fall  from  a  tan- 
gent of  its  orbit,  in  one  second  of  time,  would  be  just  equal 


192  ASTRONOMY 

.  III.  OT~ 

to  ~—.     (Art.  155.  )     In  this  expression  g,  r,  and  R,  repre- 

sent the  same  quantities  as  in  the  last  article.  The  dis- 
tance that  the  moon  actually  falls  from  a  tangent  of  its  orbit, 
in  one  second  of  time,  is  equal  to  the  versed  sine  of  the  arc  it 
describes  in  that  time,  and  the  analytical  expression  for  it  is 
found  thus  : 

Let  n-  R  represent  the  circumference  of  the  lunar  orbit,  and  if 
t  is  put  for  the  number  of  seconds  in  a  mean  revolution,  then 

*R 

—  represents  the  arc  corresponding  to  the  moon's  motion  in 

one  second  (Fig.  30),  and  as  this  so  nearly  coincides  with 
a  chord,  we  have 

ZR    •    ^    ..    1?    . 

t  t 


Hence,  we  perceive,  that  -^—-    is  the   distance   that   the 

•ion  for  the  ^  t2 

distance  the  moon  wou](j  fa]j  fr0m  the  tangent  of  its  orbit  in  one  second 

moon  falls  m  _  ° 

on*   second  of  time,  if  it  were  undisturbed  by  the  action  of  the  sun  ;  but 

of  iim«.  359 

we  can  free  it  from  such  action  by  multiplying  it  by  5^ 


as  we   shall  show  in   a  subsequent  chapter.      That  is,  the 
attraction  of  both  the  earth  and  moon,  at  the  distance  of  the 

858-** 

lunar  orbit,  is  gg^. 

But  the  attraction  of  the  earth  alone,  at  the  same  distance, 

or2 
is  ^-  ;  and  comparing  these  quantities  with  the  more  gene- 

raJ  expressions  in  Art.  156,  we  have 

gr^          358  «*R 

^      ~wr$w 

By  suppressing  the  common  denominator,  in  the  first 
couplet,  and  calling  E,  the  mass  of  the  earth,  unity,  the  pro- 
portion reduces  to 


1     :     H*    »    9+     :    -357^1 


MASSES  OF  THE  PLANETS.  193 


As  in  the  last  article,  .#=(60.  36)  r,  and  this  value  put  for  <**>•  ni> 
9,  and  reduced,  gives 

358  T*  (60.36)  3  r 


1+m    ::    g 


357-2*3 


. 
Therefore,  -     -     l+ro=  -  357.2,2 


This  fraction,  as  well  as  the  one  in  the  last  article,  can  be 
reduced  arithmetically;  but  the  operation  would  be  too 
tedious;  they  are  both  readily  reduced  by  logarithms,  by 
which  we  found  14-w=1.01333  ;  hence  m=.01333,  which 
is  very  nearly  ^th.  Laplace  says  ^jth  of  the  earth  given  by  La- 
is  the  true  mass  of  the  moon  ;  and  this  value  we  shall  use.  Place- 

THE   DENSITIES    OF   BODIES.  * 

*  . 

(173.)  The  density  of  a  body  is  only  a  comparative  term,       standard 
and  to  find  the  comparison,  some  one  body  must  be  taken  as 
the  standard  of  measure.     The  earth  is  generally  taken  for 
that  standard. 

Tt  is  an  axiom,  in  philosophy,  that  the  same  mass,  in  a 
smaller  volume,  must  be  greater  in  density;  and  larger  in 
volume,  must  be  less  in  density  ;  and,  in  short,  the  density 
must  be  directly  proportional  to  the  mass,  and  inversely  pro- 
portional to  the  volume  ;  and  if  the  earth  is  taken  for  unity 
in  ma,ss,  and  unity  in  volume,  then  it  will  be  unity  in  density 
also  ;  and  the  density  of  any  other  planetary  body  will  be  its 
mass  divided  by  Us  volume;  and  if  its  volume  is  not  given,  the 
density  may  be  found  by  the  following  proportion,  in  which 
d  represents  the  density  sought,  and  r  the  radius  of  the  body  ; 
the  radius  of  the  earth  being  unity.  The  proportion  is  drawn 
from  the  consideration  that  spheres  are  to  one  another  as  the 
cubes  of  their  radii. 

1          mass  ,     ,  mass 

•r     :     —  —     :  :     1     :    a:   hence  o=  -  . 
1  r3  r3 

9 
From  this  equation  we  readily  find  the  density  of  the  sun, 

for  we  have  its  mass  (354945),  and  its  semidiameter  111.6 
times  the  semidiameter  of  the  earth  (Art.  156)  ;  therefore  its  •«•• 
13 


194  ASTRONOMY. 


CH*P.  m.    ,      .  354945 

--    density  must  be  —=0.'ZDtl,  or  a  bttle  more  than  £ta 

•pheies  com.  (lll.Oj3 

paied  to  the  tne  density  of  the  earth. 

the  earth.          The  mass  of  Jupiter  is  332  times  that  of  the  earth,  and  its 
volume  is  1260  times  the  volume  of  the  earth  ;  therefore  the 

oon 

density  of  Jupiter  is  =0.264  ;  which  is  a  little   more 

than  the  density  of  the  sun. 

Densities      The  mass  of  the  moon  is  -^j,  and  its  volume  j1^,  therefore  its 
moon  s^c1"'  density  is  T'j  divided  by  ?^,  or  A£  =0.6533;  about  f  the  den- 
sity of  the  earth. 

From  these  examples  the  reader  will  understand  how  the 
densities  were  found,  as  expressed  in  table  III. 

GRAVITY   ON    THE   SURFACE   OP    SPHERES. 

Gravity  on      (  174.)  The  gravity  on  the  surface  of  a  sphere  depends  on 

h<5  ne'v&el  *^e  mass  an<*  volume.     The  attraction  on  the  surface  of  a 

,  »aet»,  how  sphere  is  the  same  as  if  its  whole  mass  were  collected  at  its 

foucd  center;  and  the  greater  the  distance  from  the  center  to  the 

surface,  the  less  the  attraction,  in  proportion  to  the  square  of 

the  distance  :  but  here,  as  in  the  last  article,  some  one  sphcr  » 

must  be  taken  for  the  unit,  and  we  take  the  earth,  as  before. 

The  mass  of  the  sun  is  354945,  and  the  distance  from  its 

center  to  its  surface  is  111.6  times  the  semidiameter  of  the 

«arth  ;  therefore  a  pound,  on  the  surface  of  the  earth,  is  to 

the  pressure  of  the  same  mass,  if  it  were  on  the  surface  of 

the    sun,   as    -   to  —  -,  or  as  1  to  28  nearly.     That 

is,  one  pound  on  the  surface  of  the  earth  would  be  nearly  28 
pounds  on  the  surface  of  the  sun,  if  transported  thither. 
The  mass  of  Jupiter  is  332,  and  its  radius,  compared  to 

that  of  the  earth,  is  11.1  (Art.  131);  therefore  one  pound,  on 

ooo 
the  surface  of  the  earth^  would  be  ,  or  2.  48  pounds  on 

the  surface  of  Jupiter;  and  by  the  same  principle,  we  can 
compute  the  pressure  on  the  surface  of  any  other  planet. 
Results  will  be  found  in  table  III. 


LUNAR    PERTURBATIONS.  195 


CHAPTER  IV. 


PROBLEM  OF  THE  THREE  BODIES. LUNAR  PERTURBATIONS. 

(175.)  By  the  theory  of  universal  gravitation,  every  body    CHAP  ^ 
in  the  universe  attractsevery  other  body,  in  proportion  to  its     Thelhe<>T 

J  .         of  gravity. 

mass  ;  and  inversely  as  the  square  ot  its  distance ;  but  sim- 
ple and  unexceptionable  as  the  law  really  is,  it  produces  very 
complicated  results  in  the  motions  of  the  heavenly  bodies. 

If  there  were  but  two  bodies  in  the  universe,  their  mo-     The  com" 
tions  would  be  comparatively  simple,  and  easily  t.raoed,  for  ^"(^ 
they  would  either  fall   together  or  circulate  around  each 
other  in  some  one  undeviating  curve ;   but  as  it  is,  when 
two  bodies  circulate  around  each  other,  every  other  body 
•causes  a  deviation  or  vibration  from  that  primary  curve 
that  they  would  otherwise  have. 

The  final  result  of  a  multitude  of  conflicting  motions  can- 
not be  ascertained  by  considering  the  whole  in  mass :  we  must 
take  the  disturbance  of  one  body  at  a  time,  and  settle  upon 
its  results ;  then  another  and  another,  and  so  on ;  and  the  sum 
of  the  results  will  be  the  final  result  sought. 

We,  then,  consider  two  bodies  in  motion  disturbed  by  a     The  prob- 
third  body :  and  to  find  all  its  results,  in  general  terms,   i?  *?m   of  *** 

thre*  \>o«*ies, 

the  famous  problem  of  "the  three  bodies  ;"  but  its  complete 
solution  surpasses  the  power  of  analysis,  and  the  most  skillful 
mathematician  is  obliged  to  content  himself  with  approxi- 
mations and  special  cases.  Happily,  however,  the  masses  of 
most  of  the  planets  are  so  small  in  comparison  with  the  mass 
of  the  sun,  and  their  distances  so  great,  that  their  influences 
are  insensible. 

We  shall  make  no  attempt  to  give  minute  results ;  but  we 
hope  to  show  general  principles  in  such  a  manner,  that  the 
reader  may  comprehend  the  common  inequalities  of  planetary 
motions. 

Let  m,  Fig.  35,  be  the  position  of  a  body  circulating  around       Abstram 
another  body  A,  moving  in  the  direction  PmB,  and  dig-  attnxcti<m 
turbed  by  the  attraction  of  some  distant  body  D. 


196 


ASTRONOMY 


Fig.  35.  We  now  propose  to  show  sume  of 

the  most  general  effects  of  the  ac- 
tion of  D,  wi'hout  paying  the  le  ist  re- 
gard to  quantity. 

If  A  and  ra  were  equally  at- 
tracted by  D,  and  the  attraction 
exerted  in  parallel  lines,  then  D  we  dd 
not  disturb  the  mutual  relations  af 
A  and  m  But  while  m  is  nearer  to 
D  than  A  is  to  D,  it  must  be  mor-e 
strongly  attracted,  and  let  the  lint 
mp  represent  this  excess  of  attraction. 
Decompose  this  force  (see  Nat.  Phil.) 
into  two  others,  mn  and  np,  the  first 
along  the  line  A  in,  the  other  at  right 
angles  to  it. 

The  first  is  a  lifting  force  (  called 
by  astronomers  the  radial  force), 
the  other  is  a  tang ental  force,  and  affects  the  motion  of  m.  It 
will  accelerate  the  motion  of  m,  while  acting  with  it,  from  P 
to  B;  and  retard  its  motion,  while  acting  against  it,  from  B 
to  Q. 

We  must  now  examine  the  effect,  when  the  revolving  body 
is  at  m',  a  greater  distance  from  D  than  A  is  from  D. 

Now  A  is  more  strongly  attracted  than  m',  and  the  result 
of  this  unequal  attraction  is  the  same  as  though  A  were  not 
attracted  at  all,  and  m'  attracted  the  other  way  by  a  force 
equal  to  the  difference  of  the  attractions  of  D  on  the  two 
bodies  A  and  m' .  Let  this  difference  be  represented  by  the 
line  m'p',  and  decompose  it  into  two  other  forces,  m'  n'  and 
ri p',  the  first  a  lifting  force,  the  other  the  tangcntal  force. 

The  rationale  of  this  last  position  may  not  be  perceived  b^ 
every  reader;  and  to  such  we  suggest,  that  they  conceive  A 
and  m'  joined  together  by  an  inflexible  line  A  m',  and  both 
A  and  m'  drawn  toward  7),  but  A  drawn  a  greater  dis- 
tance than  m'.  Then  it  is  plain  that  the  position  of  the  line 
A  m'  will  be  changed :  the  angle  D  Am'  will  become  greater, 
and  the  angle  C Am'  less;  that  is,  the  motion  of  m'  will  be 


LUNAR   PERTURBATIONS. 


197 


line 

sy/.igies. 


of 


accelerated  from  Q  to   C,  but  from   C  to  P  it  will  be  re-    CHAP.  iv. 
tarded. 

In  short,  the  motion  of  m  will  be  accelerated  when  moving  to-     The     du- 
ward  tlie  line  DBG,  and  retarded  while  moving  from  that  line.  turbing  ^°dy 

constantly 

That  is,  retarded  from  B  to  Q,  accelerated  from  Q  to  C,  re-  nrges  a  revoi- 
tarded  from  C  to  P,  and  again  accelerated  from  P  to  B.         ving  body  to 

If  we  conceive  A  to  be  the  earth,  m  the  moon,  arid  D  the  thl 
sun;  then  DB  C  is  called  the  line  of  the  syzigies,  a  term 
which  means  the  plane  in  which  conjunctions  and  oppositions 
take  place.  At  the  point  B  the  moon  falls  in  conjunction  with 
the  sun,  and  is  new  moon ;  at  the  point  C  it  is  in  opposition, 
or  full  moon.  Fig.  35. 

( 176.  )  Conceive  a  ring  of  matter  around 
a  sphere,  as  represented  in  Fig.  36,  and  let 
it  be  either  attached  or  detached  from  the 
sphere,  and  let  D  be  not  in  the  plane  of  the 
ring. 

From  what  was  explained  in  the  last  ar- 
ticle, the  particles  of  matter  at  m  are  con- 
stantly urged  toward  the  line  D  B  C,  and  I 
the  particles  at  m'  are  constantly  urged] 
toward  the  same  line ;  that  is,  the  at- 
traction, of  D,  on  the  ring,  has  a  tendency  to] 
diminish  its  inclination  to  the  line  DBC-\ 
and  its  position  would  be  changed  by  such 
attraction  from  what  it  would  otherwise  be ; 
and  if  the  ring  is  attached  to  the  sphere,  the  sphere  itself  will 
have  a  slight  motion  in  consequence  of  the  action  on  the  ring. 

Now  there  is,  in  fact,  a  broad  ring  attached  to  the  equator- 
ial part  of  the  earth,  giving  the  whole  a  spheroidal  form ;  and 
the  plane  of  the  equator  is  in  the  plane  of  the  ring. 

When  the  sun  or  moon  is  without  the  plane  of  this  ring,     cause    of 
that  is,  without  the  plane  of  the  equator,  their  attraction  has  nntati°n 
a  tendency  to  draw  the  plane  of  the  equator  toward  the  at- 
tracting body,  and  actually  does  so  draw  it ;  which  motion  is 
called  mitation.     How  this  motion  was  discovered,   and  its 
amount  ascertained,  will  be  explained  in  a  subsequent  chapter. 

(177.)    We  may  conceive  the  line  DBC  to  be  in  the 


ASTRONOMY. 


!!ug 


orbit, 


The  moon's 
nodes  retro* 
grade. 


Fig.  37. 


Lunar  per 
tnrbations 


Investiga- 
tion for  find- 
ing  a  general 
an  ilytical 
expression 
for  the  lunar 
perturba- 
tions. 


plane  of  the  ecliptic,  I>  the  sun,  and  the  ring  around  the  earth 
the  moon's  orbit,  inclined  to  the  plane  of  the  ecliptic  with  an 
angle  of  about  five  degrees  ;  then  when  the  sun  is  out  of  the 
plane  of  the  ring,  or  moon's  orbit,  the  action  of  the  sun  has 
a  constant  tendency  to  bring  the  moon  into  the  ecliptic,  and 
by  this  tendency  the  moon  does  fall  into  the  ecliptic  from 
either  side  sooner  than  it  otherwise  would. 

The  point  where  the  moon  falls  into  the  ecliptic  is  called 
the  moons  node;  and  by  this  external  action  of  the  sun  the 

moon  falls  into  the  ecliptic 
from  its  greatest  inclination 
before  it  describes  90°,  and 
goes  from  node  to  node  be- 
fore it  describes  180°  —  and 
hence  we  say  that  the  moon's 
nodes  fall  backward  on  the 
ecliptic.  The  rate  of  retro- 
gradation  is  19°  19'  in  a  year, 
making  a  whole  circle  in  about 
18.6  years. 

(178.)  We  are  now  pre- 
pared to  be  a  little  more  defi- 
nite, and  inquire  as  to  the 
amount  of  some  of  the  lunar 
irregularities. 

Let  S  be  the  mass  of  the 
sun,  E  that  of  the  earth,  and 
m  the  moon,  situated  at  D. 
Let  a  be  the  mean  distance 
between  the  earth  and  sun,  z 
the  distance  between  the  sun 
and  moon,  and  r  the  mean  ra- 
dius of  the  lunar  orbit.  Let 
the  moon  have  any  indefinite 
position  in  its  orbit.  ( It  is 
represented  in  the  figure  at 

c 

The  attraction  of  the  sun  on  the  earth  is  — ,  the  attrae 

a3 


LUNAR  PERTURBATIONS  199 

tion  of  the  sun  on  the  moon  is  —  ;  and  the  attraction  of  the    —  •  —  ' 

inri 

earth  and  moon,  on  the  moon,  is       ^    ,     (  Art.  156,  ) 

Let  the  line  D  B,  the  diagonal  of  the  parallelogram  A  C,  be 
the  attraction  of  the  sun  on  the  moon,  and  decompose  it  into 
the  two  forces  DA  and  D  (7;  the  first  along  the  lunar  radius 
vector,  the  other  parallel  to  SJS. 

The  two  triangles  C  D  B  and  D  S  E  are  similar,  and  give 

« 

the  proportion  a  :  z  :  :  CD  :  D  B.     But  D£  =  —  ; 

Therefore  CD  =  —^-.    By  a  similar  proportion  we  find 


Let  the  angle  SED  be  represented  by  x,  then  D  G  will 
be  expressed  by  r  cos.  #,  and  SD  G  will  be  a  right  line  nearly, 
for  the  angle  D  S  E  is,  never  greater  than  1'. 

Now  if  the  force  D  C,  which  is  parallel  to  S  E,  is  only 
equal  to  the  force  of  the  sun's  attraction  on  the  earth,  it 
will  not  disturb  the  mutual  relations  of  the  earth  and  moon. 

cr 

The  force  of  the  sun's  attraction  on  the  earth  is  —  ;  and  as  this 

must  be  less  than  the  force  of  attraction  on  the  moc-n  when 
the  moon  is  at  D,  conceive  it  represented  by  the  line  Cn,  and 
subtracted  from  (77),  will  leave  Dn  the  excess  of  the  sun's 
attraction  on  the  two  bodies,  the  earth  and  the  moon  ;  and 
this  alone  constitutes  the  disturbing  force  of  the  moon's 
motion  ; 

That  is,        Dn  =  CD—Cn  =  ^f  —  4  5  An  Mpw"~ 

Z3  a2  sion  for  the 

whole  distnr. 

Or  Dn  =  aS  {  —  —  --  Jt  the  distorting  force.     Decoin-  bing  fol 

pose  this  force  (  Dn  )  into  two  others,  Dp  and  j»n,  by  means 
of  the  right  angled  triangle  Dpn;  the  angle  pDn  being 
equal  to  DE  S,  which  we  represent  by  x. 


ASTRONOMY. 

/I  1  \ 

Whence     Dp  = :  Sa  ^  — -J  cos.  a?; 

And 

The  force  D-4,  i.e.  (  —  )  is  called  the  additions  force; 

The  radial  the  force  Dp  the  aUaiitwus  force.     The  difference  of  these 
force.  two  forces  is  called  the  radial  force ;  that  is 

Sa  (  — )  cos.  x =  the  radial  force ;  pn  is  the 

\  33       as  /  Z3 

tangental  force. 
Expression      \Vhen  the  ande  x  is  equal  to  90°,  cos.  x  =  o,  SD  —  SK 

of  the  radial 

force  at  the         __       which  values,  substituted,  give for  the  value 

quadratures.  #3 

of  the  radial  force  at  the  quadratures,  and  its  tendency  there 
is  to  increase  the  gravity  of  the  moon  to  the  earth.  When 
the  angle  x  is  zero  ( the  moon  is  in  conjunction  with  the  sun  ) 
the  cos.  x  =  1,  and  the  radial  force  becomes 

&*__Sa___rSf        S(a  —  r)       Sa 
^F"~^3~~^'or  Z3         ~~a~3' 

But  at  that  point  z  =  (  a  —  r  ),  "which  value  substituted, 
and  rejecting  the  comparatively  very  small  quantities  in  both 
numerator  and  denominator,  we  have,  for  the  radial  force  at 

2rS 
conjunction,    — — . 

When  the  angle  x  =  180°  (  the  moon  is  in  opposition  to 
the  sun  ),  cos.  x  =  —  1,  and  the  force  becomes 
Sa___  Sa  __  rS^        S        S  (a+r) 
a3       »3        z3 '    >r  a2  "         z3 

But  at  this  point  z  =  a  -(-  r,  which,  substituting  as  before, 

O        Cf 

and  we  have  for  the  radial  force  in  opposition  — --,  the  same 

expression  as  at  conjunction. 

If  we  compare  the  radial  force  at  the  syzigies  with  the  ex- 
pression for  it  at  the  quadratures,  we  shall  find  it  the  same 
in  form,  but  double  in  amount  and  opposite  in  sign,  showing 
that  it  is  opposite  in  effect. 


LUNAR   PERTURBATIONS.  201 

(179.)  As  the  radial  force  increases  the  gravity  of  the   CHAP.  rv. 
moon  to  the  earth    at  the  quadratures,  and  diminishes  it  at 

Poinlfl 

thesyzigies,  there  must  be  points  in  the  orbit  symmetrically  where  the  ra. 
situated,  in  respect  to   the   syzigies,  where  the  radial  force  dial  force  *» 
neither  increases  nor  diminishes  the  gravity,  and  of  course 
its  expression  for  those  points  must  be  zero;    and  to  find       HOW  u> 
these  points  we  must  have  the  equation  find  thenu 


os.*-        =  0     .    .     (1) 
a/  z3 

By  inspecting  the  figure  we  perceive  that  the  line  SD  Q 
is  in  value  nearly  equal  to  the  line  SE,  and  for  all  points  in 
the  orbit  we  have 

z  =  a  +  r  cos.  x  ......  (  2  ) 

Reducing  equation  (  1  ),  we  have 

(a3  —  z3  )  cos.  #  =  ra2.    .     .     .    (3) 

Cubing  (2), 


As  r  is  very  small  in  relation  to  a,  the  terms  containing  the 
powers  of  r,  after  the  first,  may  be  rejected;  we  then  have 

(03_23)  =  zp3aa  rcos.  a?.      .     .      (4) 
This  value  substituted  in  (  3  ),  and  reduced,  gives 

Result   of 

4-  3  cos.  2a?  =  1.  the     radial 

Hence    cos.  x  =  Jl    and  x  =  54°  447,    or   the   points  for<*  at  ** 

quadratures 

are  35°  16'  from  the  quadratures.  *nd  syzigieta 

This  shows  that  at  the  quadratures,  and  about  35°  on 
each  side  of  them,  the  gravity  of  the  moon  is  increased  by 
the  action  of  the  sun,  and  at  the  syzigies,  and  about  54°  on 
each  side  of  them,  the  gravity  is  diminished  ;  and  the  diminu- 
tion in  the  one  case  is  double  the  amount  of  increase  in  the     Mean   ra 
other,  and  by  the  application  of  the  differential  calculus  we  dl 
learn  that  the  mean  result,  for  the  entire  revolution,  is  a  dimi- 

nution whose    analytical  expression  is   ^—  -  ,   an  expression 
which  holds  a  very  prominent  place  in  the  lunar  theory;  the 


202  ASTRONOMY. 


.  result  of  which  we  have  used  in  Art.  171,  and  there  stated  it 
to  be  ^1  g  th  part  of  the  force  that  retained  the  moon  in  its 
orbit. 

Value    of      But  how  do  we  know  this  to  be  its  numerical  value,  is  a 
dilimeafor^r  verv  8eri°us  inquiry  of  the  critical  student? 
fold    b°W      The  force  that  retains  tlie  moon  in  its  orbit  is  .E  +  m. 
(  Art.  156  )  ;  and  if  the  radial  force  can  be  rendered  homoge- 
neous with    this,  some  numerical  ratio   must  exist   between 
them.     Let  x  represent  that  ratio,  and  we  must  find  dome 
numerical  value  for  x  to  satisfy  the  following  equation  : 
rS_          E+m 


Therefore         r  = 

calling  JS=  1,  m  =  ^  (Art.  172),  or  E  -\-  m  is  1.013. 
S  =  354945  (  Art.  169  ),  and  the  relation  between  th* 
mean  distance  to  the  sun,  and  the  mean  radius  of  the  lunai 
orbit,  is  397.3,*  therefore 


or  the  coefficient  to  x,  in  equation  (  A  ),  is  one  three  hundredth 
and  fifty-eighth  part  of  the  force  which  retains  the  moon  in  its 
orbit. 

General  ef.  (180.)  The  mean  radial  force  causes  the  moon  to  circu- 
diai  force.  ^a^e  a^  3^8^  Part  grater  distance  from  the  earth  than  it 
otherwise  would  have,  and  its  periodical  revolution  is  in- 
creased by  its  179th  part  ;  but  this  would  cause  no  variation 
or  irregularity  in  its  distance  or  angular  motion,  provided  its 
orbit  were  circular,  and  the  earth  and  moon  always  at  the 
name  mean  distance  from  the  sun. 

The  radial      But  we  perceive  the  expression  ^-^  contains   two  variable 

force    varia. 

w«.  quantities,  r  and  a,  which  are  not  always  the  same  in  value  ; 

and,  therefore,  the  value  of  the  expression  itself  must  be  va- 

*  This  relation  is  found  by  dividing  the  horizontal  parallax  of  the 
moon,  56'  57",  by  the  horizontal  parallax  of  the  sun,  8".6. 


iJNAR  PERTURBATIONS.  203 

riable  ;   and  it  will  be  least  when  the  earth  is  at  the  greatest   CHAP.  tv. 

distance  from  the  sun,  and,  of  course,  the  moon's  motion  will 

then  be  increased.     But  the  earth's  variable  distance  from 

the  sun  depends  on  the  eccentricity  of  the  earth's  orbit;  and    The  annu. 

hence  we  perceive  that  the  same  cause  which  affects  the  ap-  al   eiuatlon 

r  r     of  the  moon'* 

parent  solar  motion,  affects  also  the  motion  of  the  moon,  and  motion. 
gives  rise  to  an  equation  called  the  annual  equation*  of  the 
moon's  motion.     It  amounts  to  11'  in  its  maximum,  and  va- 
ries by  the  same  law  as  the  equation  of  the  sun's  center. 

(  181.)  If  we  take  the  general  expressions  for  the  radial      A  general 

^  expression 

force,    Sa(—  —  -L)  cos.  x  —  -,  and  banish  the  letter  z  J"  theradlal 

\23          a3'  23  force  at  any 

/.  ..   i  /.i  .  point  of  tha 

from  it  by  means  or  the  equation  moon's  orbit. 

2    =  a   4-  r  cos.  x 

Or,  z3  =  a3  +  3a2  r  cos.  x, 

(  neglecting  the  powers  of  r  )  and  we  shall  have, 

rS  (3  cos.  2x  —  1) 

«3 

for  an  expression  of  the  radial  force  corresponding  to  any 
angle  x  from  the  syzigy. 

If  we  take  the  general  expression  for  the  line  pn,  the  tan- 
gental  force,  and  banish  2,  as  before,  we  have, 

3rs  cos.  x  sin.  x 
tangental  force  =  -  .• 


By  doubling  numerator  and  denominator,  this  fraction  can    Expression 
(2  cos.  x  sin.  x\ 


take  the  following  form  :  for  the  tan' 

gental  force. 


But,  by  trigonometry,  2  cos.  x  sin.  x  =  sin.  2#, 

mi       f       .1  .  i  /.  %rs  sin.  2x 

Ineretore  the  tangental  force  =  -  ^  --  . 

This  expression  vanishes  when  x  =  o  and  x  =  90°  ;  for  then    its  vanish 
gin.  2x  —  sin.  180  =  0.     Hence  the   tangental   force   van-  in*P°inM 
xshes  at  the  syzigies  and  quadratures,  attains  its  maximum 

»  This  is  equation  I,  in  the  Lunar  Tables. 


204 


ASTRONOMY. 


The 


tan- 
°rCe 


when  the 
earth    is    in 
perigee. 


Application 

force*  to  an  on 
elliptical  o: 


CHAP.  iv.    value  at  the  octants,  and  varies  as  the  sine  of  the  double  angular 
distance  of  the  moon  from  the  sun. 

The  mean  maximum  for  this  force  must  be  determined  by 
observation.  It  is  known  by  the  name  of  variation,  and  by 
mere  inspection  we  can  see  that  its  amount  must  correspond 
to  the  variations  of  r  and  of  a3.  Hence,  to  obtain  the  moon's 
place,  we  must  have  correction  on  correction. 

The  variation  amounts  to  about  35'.  It  increases  the  ve- 
locity of  the  moon  from  the  quadratures  to  the  syzigies,  and 
diminishes  it  from  the  syzigies  to  the  quadratures ;  hence,  in 
consequence  of  the  variation,  the  velocity  of  the  moon  is 
greatest  at  the  syzigies,  and  least  at  the  quadratures. 

(182.)  Let  us  now  examine  the  effect  of  the  radial  force 
lunar  orbit,  considered  as  elliptical. 

Let  SE(Y\g.  38)  be  at  right 
angles  to  A  JB,  the  greater  axis 
of  the  lunar  orbit,  and  conceive 
A  C  B  to  represent  the  orbit  that 
the  moon  would  take  if  it  were 
undisturbed  by  the  sun. 

But  when  the  moon  comes 
round  to  its  perigee  at  A,  it  is  in 
one  of  its  quadratures,  and  the 
radial  force  then  increases  the 
gravity  of  the  moon  toward  the 

earth  by  the  expression  — .    But 

here  r  is  less  than  its  mean  value, 
and  the  expression  is  less  than  its 
mean,  and  therefore  the  moon  is 
not  crowded  so  near  the  earth  ag 
it  otherwise  would  be,  and,  of 
course,  at  this  point  the  moon 
will  run  farther  from  the  earth. 

At  the  point  C,  the  radial  force  tends  to  increase  the  dis- 
tance between  the  earth  and  moon,  and  (o  widen  the  orbit. 
when  the      Wh<m  the  moon  passes  round  to  B,  the  radial  force  again 
force  increases  the  gravity  of  the  moon,  and  r,  in  the  expression 


LUNAR  PERTURBATIONS  205 

T8      .  CHAP    IV. 

"£,  w  greater  than  its  mean  value ;  and,  of  course,  crowds  the 

"  decreases 

moon  nearer  to  the  earth  than  it  otherwise  would  go,;  and  the  e«cent" 
thus  we  perceive  that  the  action  of  the  radial  force  on  an  el-  Taf  ellipse " 
liptical  orbit  has  a  tendency  to  decrease  the  eccentricity  of  the 
ellipse,  when  the  sun  is  at  right  angles  to  its  greater  axis. 

(  183.)  Now  conceive  the  sun  to  be  in  a  line,  or  nearly  in 
a  line,  with  the  longer  axis  of  the  lunar  orbit,  as  represented 
in  Fig.  39.  Fig.  39. 

The  radial  force  at  the  quadratures,  I 
C  and  D,  has  a  tendency  to  press  in 
the  orbit,  or  narrow  it.  At  the  point 
A,  the  tendency,  it  is  true,  is  to  in- 
crease the  distance  between  the  earth 
and  moon;  but  that  tendency  is  not 
so  strong  as  it  would  be  if  the  moon 
were  at  its  mean  distance  from  the! 
earth. 

The  tendency  at  B  is  to  increase  I 
the  distance,  and  it  is  a  tendency 
greater  than  the  medium.  That  is, 
the  tendency  at  A  is  less  than  the 
medium;  at  B,  greater  than  the  me- 
dium; and  at  C  and  D,  the  com-| 
pressed  parts  of  the  orbit,  the  ten- 
dency is  to  a  still  greater  compres- 
sion; therefore,  the  entire  action 
the  radial  force  is  to  increase  the  ec- 
centricity of  the  lunar  orbit,  when  the\ 
sun  is  in  line,  or  nearly  in  line,  with 
the  longer  axis. 

Thus,  we  perceive,  that  under  the  disturbing  action  of  the 
sun,  the  eccentricity  of  the  moon's  orbit  must  be  in  a  state 
of  perpetual  change,  now  more,  now  less,  than  its  mean  state. 

Corresponding  with  this  change  of  eccentricity  there  must 
be  changes  in  the  lunar  motion ;  and  to  keep  account  of  it, 
and  allow  for  it,  astronomers  have  formed  a  table  called 

EVECTION. 


206 


ASTRONOMY. 


CMMlH'        (1?4.)  Now  let  us  examine  the  effect  of  the  radial  force 
Effect  of  on  the  position  of  the  lunar  apogee. 

the        radial 

^  Let  E  (Fig.  40),  be  the  earth,  and, 

for  the  sake  of  simplicity,  we  conceive 
the  earth  to  be  stationary,  and  the 
sun  and  moon  both  to  revolve  about 
it  with  their  apparent  angular  veloci- 
ties ;  the  moon  in  the  orbit  A  C  JB, 
and  in  the  direction  A  C  B-,  the 
sun  in  a  distant  orbit,  part  of  which 
is  represented  by  S  S'. 

Let  A  B  be  the  greater  axis  of  the 
moon's  orbit,  in  its  natural  position, 
pr  as  it  would  be  if  undisturbed  by 
the  sun ;  and  being  undisturbed,  the 
perigee  and  apogee  would  remain  con- 
stant at  the  points  A  and  B\  and  the 
time  from  A  to  B,  or  from  B  to  A, 
would  be  just  equal  to  the  mean  time 
of  half  a  revolution,  as  explained  in 
a  former  part  of  this  work. 

Now  let  us  conceive  the  sun  to  be 
in  its  orbit  at  S,  then  the  moon  will 
be  in  the  syzigy  when  it  comes  round 

to  s,  and  as  the  radial  force  at  that  point  tends  to  increase 
the  distance  between  the  earth  and  the  moon,  the  apogee  will 
take  place  at  s,  or  between  s  and  B;  and  it  is  evident  that 
the  apogee  in  that  case  would  recede  or  run  back.  But  at 
revolution  of  the  moon,  in  a  little  more  than  twenty- 
lunar  orbit  is  seven  days,  the  sun  at  that  time  will,  apparently,  have  moved 
follow  the  to  Sr  about  twenty-seven  degrees.  Now  the  syzigy  will  take 
snn.  place  at  s ',  and  the  greatest  distance  between  the  earth  and 

moon  will  now  be  between  B  and  *',  that  is,  the  apogee  will 
advance,  in  one  revolution,  from  near  *  to  near  s';  and  thus, 
in  general,  the  longer  axis  of  the  moon's  orbit  is  strongly  in- 
clined to  follow  the  sun ;  and  this  is  the  source  of  its  pro- 
gressive motion.  It  makes  a  revolution  in  32321  days; 
but  its  motion  is  very  irregular,  for,  as  we  have  just  seen, 


Retrograde 
motion  ofthe 
perigee  and 
apogee. 


The  major 
axis    of   the  *ne 


LUNAR  PERTURBATIONS  207 

when  ,the  line  which  joins  the  earth  and  sun  makes  a  very  CHAP,  iv 
acute  angle  with  the  longer  axis  of  the  lunar  orbit,  and  is  ap- 
proaching that  axis,  the  motion  of  the  apogee  and  perigee  is 
retrograde;  but,  all  of  a  sudden,  when  the  sun  passes  the 
longer  axis  of  the  lunar  orbit,  the  motion  of  the  apogee  be- 
comes direct,  and  moves  with  considerable  rapidity. 

When  the  sun  is  at  right  angles  to  the  major  axis  of  the  Under  what 
moon's  orbit,  the  tendency  of  the  radial  force  is  to  diminish  P°sition  of 
the  eccentricity  of  the  orbit,  but  it  has  no  tendency  to  change  innar  perigee 

the  position  of  the  axis.  remains  «ta- 

From  this  investigation  it  follows,  that  when  the  sun  has  *" 
just  passed  the  greater  axis  of  the  lunar  orbit,  the  interval 
from  apogee  to  apogee,  or  from  perigee  to  perigee,  will  be 
greater  than  a  revolution.  Just  before  the  sun  arrives  at  the 
position  of  the  longer  axis,  the  time  from  one  apogee  to  an- 
other is  less  than  a  revolution;  and  when  the  sun  is  at 

nt  angles  to  the  longer  axis,  the  time  is  just  equal  to  a 
^."olution  in  longitude. 

(185.)  By  comparing  eclipses  of  the  moon,  observed  by        Ancient 
the  ancient  Egyptians  and  Chaldeans,  with  those  of  more  e 
modern  times,  Dr.  Halley,  and  other  astronomers,  concluded  modem 
that  the  periodic  time  of  the  moon  is  now  a  little  shorter  sedation, 
than  at  those  remote  periods;  and  to  make  these  extreme 
observations  agree  with  modern  ones,  it  became  necessary 
to  conceive  the  moon's  mean  motion  to  be  accelerated  about 
11  seconds  per  century. 

For  a  long  time  this  fact  seriously  perplexed  astronomers :      Ti*   « 
some  were  for  condemning  the  theory  of  gravity  as  insuffi-  *nt" 
cient  to  explain  the  cause  of  the  lunar  perturbations,  while 
others  were  for  rejecting  the  facts,  although  as  well  estab- 
lished as  any  mere  historical  facts  could  be. 

In  this  dilemma,  says  Herschel,  "Laplace  stepped  in  to 
rescue  physical  astronomy  from  reproach  by  pointing  out  the 
real  cause  of  the  phenomenon  in  question." 

Although  this  subject  troubled  the  greatest  philosophers 
of  the  past  age  —  the  greatest  mathematical  philosophers  the 
world  ever  saw  —  the  problem  is  quite  simple,  now  the  solu- 
tion is  pointed  out,  and  we  are  sure  that  every  reader  of  or- 


208  ASTRONOMY 

CHAP-  1V-    dinary  capacity  can  understand  it,  provided  he  gives  iiis  se- 
rious attention  to  the  subject. 
A  summary      rpjie   secu]ar  acceleration   of  the  moon's  mean  motion  is 

statement  of 

the  cause,      caused  by  a  small  change  in  the  mean  value  of  the  radial  force, 
occasioned  by  a  change  in  the  eccentricity  of  the  earth's  orbit. 

The  expression  —  -  is   the  mean  radial  force  of  the  sun 

acting  on  the  moon's  orbit,  dilating  it  and  increasing  the 
time  of  the  lunar  revolution. 

When  the      jf  ^e  earth's  orbit  had  no  eccentricity,  2a3,  the  denomina- 
tion^ "is   in-  ^or  °f  tne  fraction,  would  always  have  the  same  value,  and 
creased.        then  regarding  the  numerator  as  constant,  there  would  be  no 
variation  of  the  moon's  motion  arising  from  this  cause.     But 
in  consequence  of  the  earth  and  moon  moving  toward  the 
apogee    of    the   earth's    orbit,  a,    of    course,    a3    becomes 
greater,  and  the  value  of  the  radial  force  becomes  less  than 
its  mean  value,  and  in  consequence  of  this,  the  moon's  3 
tion  is  increased.     And  when  the  earth  and  moon  move  «/.? 
Wh«n  di-  ward  the  earth's  perigee,  a  and  a3  become  less,  and   the 
value  of  the  radial  force  becomes  greater  than  its  mean ;  the 
moon's  orbit  is  dilated  to  excess,  and  its  motion  is  diminished ; 
rhe    ex.  and  the  orbit  is  more  dilated  when  the  earth  is  in  perigee  than  it 
pression   for  ^  coniracie^  when  the  earth  is  in  apogee.     In  other  words,  the 

the  mean  ra-  .      . 

dial  force  is  mean  dilatation  of  the  lunar  orbit  is  greater,  and  the  mean 
not  the  woe  motion  of  the  moon  less,  in  proportion  as  the  earth's  orbit  is 
more  eccentric. 

The  less  the  value  of  —  the  greater  is  the  moon's  mean 

motion,  and  that  value  is  least  when  a  is  greatest.  But  a 
would  have  no  variation  of  value  if  the  earth's  orbit  were 
circular. 

The  earth's  orbit,  however,  is  eccentric,  and  in  the  course 
of  a  year  the  value  of  the  radial  force  is  exactly  expressed 

by  ~—  only  at  two  instants  of  time,  when  the  earth  passes 

the  extremities  of  the  shorter  axis  of  its  orbit.  At  all 
other  times  a  is  either  greater  or  less  than  its  mean 
value,  and  the  variations  are  equal  on  each  side  of  it ;  that 


LUNAR   PERTURBATIONS.  209 

ii,  a  baeomes  (a  —  d)  or  (a-|~ef),  and  the  radial  force  is  CHAP.  IT. 
really 

rS  rS 

^OF   2 

which  expressions  correspond  to  equal  distances  on  each  side     The  trae 
of  the  mean  distance,  and  d  may  have  all  values  from  0  to  of  the 
a  e,   the   eccentricity.     The  mean  value  of  the  radial  force  *>«•• 
corresponding  to  the  whole  yeart  is  equal  to 


-(-* 

2\(a 


i's     |    *rS  V 

i— d)*^(a+d)*/' 

Or  r8(      l         4-     l       } 

4V(a— rf)»^(5+5j>/' 

« 

But   this  expression   is  always  greater   than   - — ,  except    The  meaB 

2a3  value  of  the 

when  d  =  0 ;  then  it  is  the  same,  as  any  algebraist  can  verify.  radial   ' 
Hence  the  mean  radial  force  for  the  whole  year  is  greater  Of  ail  when 
as  the  earth's  orbit  is  more  eccentric,  and  it  will  be  least  of  the.    earlh>i 
all  when  that  orbit  becomes  a  circle ;  and  then,  and  then  circie. 

only,  it  will  be  accurately  represented  by  ^--. 

But  when  the  radial  force  is  least,  the  mean  motion  must 
be  greatest,  and  that  force  is  less  and  less  as  the  eccentricity 
of  the  earth's  orbit  becomes  less  and  less;  and  corresponding 
thereto  the  moon's  motion  becomes  greater  and  greater,  as 
has  been  the  case  for  more  than  4000  years. 

(  186.  )  The  mean  distance  between  the  earth  and  sun  re-     The  can«» 
mains  constant.     It  must  be  so  from  the  nature  of  motion,  *f  '^JJJUJ! 
force,    action,  and   reaction ;    but  by  the  attraction  of  the  city  of  th» 
planets  the  eccentricity  of  the  earth's  orbit  is  in  a  state  of  per-  earth'*  orblt 
petual  change :  the  change,  however,  is  excessively  slow.     From 
the  earliest  ages  the  eccentricity  of  the  orbit  has  been  dimin- 
ishing ;  and  this  diminution  will  probably  continue  until  it  is 
annihilated  altogether,  and  the  orbit  becomes  a  circle ;  after 
which  it  will  open  out  in  another  direction,  again  become  ec- 
centric, and  increase  in  eccentricity  to  a  certain  moderate 
amount,  and  then  again  decrease. 
14 


210  ASTRONOMY. 

CHAP.  IV.       The  period  for  these  vibrations,  "  though  calculable,  has  never 
The    im.  feen  calculated  further  than  to  satisfy  us  that  it  is  not  to  be 
™nespo™d  reckoned  by  hundreds  or  even  by  thousands  of  years."     It  is  a 
ding  to  these  period  so  long  that  the  history  of  astronomy,  and  of  the  whole 
human  race,  is  but  a  point  in  comparison. 

The  moon's  mean  motion  will  continue  to  increase  until  the 
earth's  orbit  becomes  a  circle;  after  which  it  will  again  decrease, 
corresponding  with  the  increase  of  a  new  eccentricity. 

( 187' )  For  the  sake  of  simplicity,  we  have  thus  far  con- 
lunar    orbit  sidered  the  moon's  orbit  to  be  in    the  same  plane  as  the 
account  int°  eartn's  orki* »  kut  this  is  not  true ;  the  mean  inclination  of  the 
lunar  orbit  to  the  ecliptic  is  5°  8',  varying  about  9'  each  way, 
according  to  the  position  of  the  sun. 

Owing  to  this  inclination  of  the  lunar  orbit,  the  expressions 
which  we  have  obtained  for  the  tangental  force  need  cor- 
rection, by  multiplying  them  by  the  cosine  of  the  inclination ; 
and  for  the  effect  of  the  same  forces  in  a  perpendicular 
direction  to  the  moon's  longitude,  multiply  them  by  the  sine 
of  the  inclination  of  the  orbit. 

The  position  of  the  moon's  orbit,  in  relation  to  the  sun,  is 
strictly  analogous  to  the  ring  in  relation  to  the  disturbing 
body  D  (Art.  176) ;  the  sun  is  constantly  urging  the  moon 
into  the  plane  of  the  ecliptic,  which  has  a  constant  tendency 
to  diminish  the  inclination  of  the  lunar  orbit  (  except  when 
the  sun  is  in  the  positions  of  the  moon's  nodes) ;  and  this  con- 
stant force  urging  the  moon  to  the  ecliptic,  causes  the  moon's 
nodes  to  retrograde. 

We  conclude  this  chapter  by  a  brief  summary  of  the  prin- 
cipal causes  which  affect  the  moon's  motion. 
A  summary      j>  The  eccentricity  of  the  earth's  orbit ;  which  gives  rise  to 

itatementof   ,,  \  f   ,,  .      ,          .,      , 

the  lunar  ir-  *"e  annual  equation  of  the  moon  in  longitude. 

2.  The  eccentricity  of  the  lunar  orbit ;  producing  the  equa- 
tion of  the  center. 

3.  The  tangental  force;  giving  rise  to  the  equation  called 
variation. 

4.  The  position  of  the  sun  in  respect  to  the  greater  axil 
of  the  lunar  orbit ;  giving  rise  to  the  inequality  called  evettwn. 

5.  The  inclination  of  the  moon's  orbit. 


THE    TIDES.  211 

6.  The  combination  of  the  first  cause,  when  differing  from  CHAP.  IV. 
its  mean  state,  augments  or  diminishes  the  result  of  every 

other  —  thus  making  many  additional  small  equations 

7.  The  ellipsoidal  form  of  the  earth. 


CHAPTER    V. 

THE    TIDES. 

(  188.  )  THE  alternate  rise  and  fall  of  the  surface  of  the  CHAT.  v. 
sea,  as  observed  at  all  places  directly  connected  with  the  Definition 
waters  of  the  ocean,  is  called  tide  ;  and  before  its  cause  was  of  the  ternt 

tide. 

definitely  known,  it  was  recognized  as  having  some  hidden  and 
mysterious  connection  with  the  moon,  for  it  rose  and  fell  twice    Connection 
in  every  lunar  day.     High  water  and  low  water  had  no  con-  WIth       th* 

J  *  °  moon. 

nection  with  the  hour  of  the  day,  but  it  always  occurred  in 
about  such  an  interval  of  time  after  the  moon  had  passed  the 
meridian. 

When  the  sun  and  moon  were  in  conjunction,  or  in  opposi-    HigUtidei. 
tion,  the  tides  were  observed  to  be  higher  than  usual. 

When  the  moon  was  nearest  the  earth,  in  her  perigee,  other 
circumstances  being  equal,  the  tides  were  observed  to  be 
higher  than  when,  under  the  same  circumstances,  the  moon 
was  in  her  apogee. 

The  space  of  time  from  one  tide  to  another,  or  from 
high  water  to  high  water  (  when  undisturbed  by  wind  ),  is 
12  hours  and  about  24  minutes,  thus  making  two  tides  in  one 
lunar  day  ;  showing  high  water  on  opposite  sides  of  the  earth 
at  the  same  time. 

The  declination  of  the  moon,  also,  has  a  very  sensible  influ-     Tide«    at- 
ence  on  the  tides.     When  the  declination  is  high  in  the  north,  [ect,ed  b?  the 

1  decimation 

the  tide  in  the  northern  hemisphere,  which  is  next  to  the  moon,  Of  the  moon. 
is  greater  than  the  opposite  tide  ;  and  when  the  declination  of 
the  moon  is  south,  the  tide  opposite  to  the  moon  is  greatest.      A  difficulty 
It  is  considered  mysterious,  by  most  persons,  that  the  moon  * 


by  its  attraction  should  be  able  to  raise  a  tide  on  the  opposite  wagoner 
side  of  the  earth. 


212 


ASTRONOMY. 


CHAP.  V. 


The     true 
can** 


Fig.  41. 


A  summary 
illustration 
of  the  tides. 


That  the  moon  should  attract  the  water  on  the  side  of  the 
earth  next  to  her,  and  thereby  raise  a  tide,  seems  rational  and 
natural,  but  that  the  same  simple  action  also  raises  the  oppo- 
site tide,  is  not  readily  admitted ;  and,  in  the  absence  of  clear 
illustration,  it  has  often  excited  mental  rebellion  —  and  not  a 
few  popular  lecturers  have  attempted  explanations  from  false 
and  inadequate  causes. 

But  the  true  cause  is  the  sun  and  moon's  attraction ;  and 
until  this  is  clearly  and  decidedly 
understood  —  not  merely  assented 
to,  but  fully  comprehended' —  it  ia 
impossible  to  understand  the  com- 
mon results  of  the  theory  of  gra- 
vity, which  are  constantly  exem- 
plified in  the  solar  system. 

We  now  give  a  rude,  but  strik- 
ing, and,  we  hope,  a  satisfactory 
explanation. 

Conceive  the  frame-work  of  the 
earth  to  be  an  inflexible  solid,  as  it 
really  is,  composed  of  rock,  and  in- 
capable of  changing  its  form  under 
any  degree  of  attraction  ;  conceive 
also  that  this  solid  protuberates 
out  of  the  sea,  at  opposite  points  of 
the  earth,  at  A  and  B,  as  repre- 
sented in  Fig.  41,  A  being  on  the 
side  of  the  earth  next  to  the  moon, 
m,  and  B  opposite  to  it.  Now  in 
connection  with  this  solid  con- 
ceive a  great  portion  of  the  earth 
to  be  composed  of  water,  whoso 
particles  are  inert,  but  readily 
move  among  themselves. 

The  solid  A  B  cannot  expand  under  the  moon's  attrac- 
tion, and  if  it  move,  the  whole  mass  moves  together,  in  virtue 
of  the  moon's  attraction  on  its  center  of  gravity.  But  the 
particles  of  water  at  a,  being  free  to  move,  and  being  under  a 


THE    TIDES.  213 

more  powerful  attraction  than  the  solid,  rise   toward  A,  pro-     CHAP.  ?. 
ducing  a  tide. 

The  particles  of  water  at  b  being  less  attracted  toward  m 
than  the  solid,  will  not  move  toward  m  as  fast  as  the 
solid,  and  being  inert,  they  will  be,  as  it  were,  left  behind. 
The  solid  is  drawn  toward  the  moon  more  powerfully  than  the 
particles  of  water  at  b,  and  sinks  in  part  into  the  water,  but 
the  observer  at  B,  of  course,  conceives  it  the  water  rising  up 
on  the  shore  (which  in  effect  it  is),  thereby  producing  a 
tide. 

(  189. )  The  mathematical  astronomer  perceives  a  strict  Analogy 
analogy  between  the  analytical  expressions  for  the  tides  and  i*n^e^rtnr. 
the  expressions  for  the  perturbations  of  the  lunar  motion.  bations  and 

What  we  have  called  the  radial  force,  in  treating  of  the  the  P61*"**. 

*  tions  of    th« 

lunar  irregularities,  is  the  same  in  its  nature  as  the  force  that  ocean 
raises  the  tides ;  the  tide  force  is  a  radial  force,  which  dimi- 
nishes the  pressure  of  the  water  toward  the  center  of  the 
earth  under  and  opposite  to  the  moon,  in  the  same  manner  as 
the  radial  force  diminishes  the  gravity  of  the  moon  toward 
the  earth  in  her  syzigies. 

In  Art.  179  we  found  that  the  radial  force  for  the  moon,  at     The  radiaj 

2r£  force   as  ap. 

the  syzigies,  is  expressed  by  — — ;  in  which  expression  S  is  Plied  to  ^ 

Q  moon. 

the  mass  of  the  sun,  a  its  distance  from  the  earth,  and  r  the 
radius  of  the  lunar  orbit. 

The  same  expression  is  true  for  the  tides,  if  we  change  S  to     Convert<Mi 

i  into    an    ex. 

m,  the  mass  ot  the  moon,  and  conceive  a  to  represent  the  dis-  pression   ft* 
tance  to  the  moon,  and  r  the  radius  of  the  earth.     For  the  the  tides 

tides,  then,  we  have  — — ,  and  as  the  numerator  is  always  con- 
stant, the  variation  of  the  tides  must  correspond  to  the  cube 
of  the  inverse  distance  to  the  moon. 

(  190.)  The  sun's  attraction  on  the  earth  is  vastly  greater      Sun'8  •* 
than  that  of  the  moon ;  but  by  reason  of  the  great  distance  $idered. 
to  the  sun,  that  body  attracts  every  part  of  the  earth  nearly 
alike,  and,  therefore,  it  has  much  less  influence  in  raising  a 
tide  than  the  moon. 


214  ASTRONOMY. 

CHAP,  v.        From    a   long  course  of  observations  made  at  Brest,  in 

Observations  France,  it  has  been  decided  that  the  medium  high   tides, 

when    the   sun  and   moon  act   together   in  the  syzigies,  is 

19.317   feet;  and  when  they  act   against  each  other  (the 

moon  in  quadrature),  the  tides  are  only  9.151  feet.     Hence 

Compara-  tne  e{gcacy  Of  the  moon,  in  producing  the  tides,  is  to  that 

ceTofTheTun  of  the  sun,  as  the  number  14.23  to  5.08. 

•ndmoon,         Among  the  islands  in  the  Pacific  ocean,  observations  give 

the  proportion  of  5  to  2.2,  for  the  relative  influences  of  these 

two  bodies  ;  and,  as  this  locality  is  more  favorable  to  accu- 

racy than  that  of  Brest,  it  is  the  proportion  generally  taken. 

Having  the  relative  influences  of  two  bodies  in  raising  the 

tides,  we  have  the  relative  masses  of  those  two  bodies,  pro- 

vided they  are  at  the  same  distance.     But  by  the  expression 

for  the  tides,  as  we  have  just  seen,  the  variation  for  distance 

corresponds  with  the  inverse  cube  of  the  distance,  and  the  dis- 

tance to  the  sun  is  397.2  times  the  mean  distance  to  the 

moon.     Hence,  to  have  the  influence  of  the  moon  on  the 

tides,  when  that  body  is  removed  to  the  distance  of  the  sun, 

we  must  divide  its  observed  influence  by  the  cube  of  397.2. 

Mass  of  the  rpnaf.  jg  ^Q  mass  of  the  moon  is.  to  the  mass  of  the  sun,  as 

moon     com. 

the  number  --    to  the  number  2.2. 


In  all  preceding  computations  we  have  called  the  mass  of 
the  earth  unity,  and  in  relation  thereto,  the  mass  of  the  sun  is 
354945  (  Art.  169).  Let  us  represent  the  mass  of  the  moon 
by  m,  then  we  have  the  following  proportion  : 

m  i  354945  :  :        5        :  2.2. 


This  proportion  makes  the  mass  of  the  moon  a  little  less  than 
j\  ;  but  I  have  little  confidence  in  the  accuracy  of  the  result, 
as  the  data,  from  their  very  nature,  must  be  vague  and  in- 
definite. 

The  timei  (  191.)  The  time  of  high  water  at  any  given  point  is  not 
ter  afferent  commonly  at  the  time  the  moon  is  on  the  meridian,  but  two 
in  aifferent  or  three  hours  after,  owing  to  the  inertia  of  the  water  ;  and 
localities.  pjaceSj  not  far  from  each  other,  have  high  water  at  very  dif- 


THE  TIDES.  215 

ferent  times  on  the  same  day,  according  to  the  distance  and    CHAN  v 
direction  that  the  tide  wave  has  to  undulate  from  the  main 
ocean. 

The  interval  between  the  meridian  passage  of  the  moon 
and  the  time  of  high  water,  is  nearly  constant  at  the  same 
place.  It  is  about  fifteen  minutes  less  at  the  syzigies  than 
at  the  quadratures;  but  whatever  the  mean  interval  is  at 
any  place,  it  is  called  the  establishment  of  the  port. 

It  is  high  water  at  Hudson,  on  the  Hudson  river,  before  The  tides 
it  is  high  water  at  New  York,  on  the  same  day;  but  the  tide  gtantjy cease 
wave  that  makes  high  water  one  day  at  Hudson,  made  high  on  the  remo- 
water  at  New  York  the  day  before ;  and  the  tide  waves  that  ™°  their 
make  high  water  now,  were,  probably,  raised  in  the  ocean 
several  days  ago ;  and  the  tides  would  not  instantly  cease  on 
^  annihilation  of  the  sun  and  moon. 

The  actual  rise  of  the  tide  is  very  different  in  different    Tides  ver* 
places,  being  greatly  influenced  by  local  circumstances,  such  ™ducb 
as  the  distance  and  direction  to  the  main  ocean,  the  shape  circum- 
of  the  bay  or  river,  &c.,  &c.  stance*. 

In  the  Bay  of  Fundy  the  tide  is  sometimes  fifty  and  sixty 
feet ;  in  the  Pacific  ocean  it  is  about  two  feet ;  and  in  some 
places  in  the  West  Indies,  it  is  scarcely  fifteen  inches.  In 
inland  seas  and  lakes  there  are  no  tides,  because  the  moon's 
attraction  is  equal  over  their  whole  extent  of  surface. 

The  following  table  shows  the  hight  of  the  tides  at  the 
most  important  points  along  the  coast  of  the  United  States, 
as  ascertained  by  recent  observation. 

Feet. 

Annapolis  (Bay  of  Fundy) 60 

Apple  River, 50 

Chicneito  Bay  (north  part  of  the  Bay  of  Fundy) 60 

Passamaquoddy  River, 25 

Penobscot  River, 10 

Boston, 11 

Providence, R.  I., «...     5 

New  Bedford, 5 

New  Haven, 8 

New  York, 5 

Cape  May, 6 

Cape  Henry, 4% 


216  ASTRO     OMT. 

CHAPTER   VI. 

PLANETARY   PERTURBATIONS. 

V""  ™-       (  192.)  The  perturbations  of  a  planet,  produced  by  the  at- 

Pianetary  tractions  of  another  planet,  are  precisely  analogous  to  the  per- 

pertnrba-       turbations  of  the  moon,  produced  by  the  action  of  the   sun. 

tions  aaaio-  The  disturbing  forces  are  of  the  same  kind,  and  they  are 

subject  to  similar  variations  from  precisely  the  same  causes. 

But  the  amount  of  the    disturbances  is,  in  most  cases,  very 

trifling,  on  account  of  the  small  mass  of  the  disturbing  planet 

compared  with  the  mass  of  the  sun,  or  its  great  distance  from 

«he  body  disturbed. 

Action  and      AS  action  and  reaction  are  everywhere  equal,  the  planets 

ong  the  plan"  Mutually  disturb  each  other,  and  if  one  is  accelerated  in  its 

•u    recipro-  motion,  the  other  must  be  retarded  ;  if  the  tendency  of  one  to- 

ward the  sun  is  diminished,  that  of  the  other  must  be  increased. 

Examine  Fig.  23,  and  conceive  V,  Venus,  to  be  disturbed 

by  the  attraction  of  the  earth  at  E,  and  if  the  motion  of  the 

planets  is  in  the  direction  of   VJB,  it  is  perfectly  clear  that 

Venus  will  be  accelerated  by  the  earth,  and  the  earth  will  be 

retarded  by  Venus. 

One  planet  But  Venus  will  be  more  accelerated  in  its  motion  thaw  the 
•a  while  an^  eart>n  wiW  ^e  retarded,  for  the  disturbance  at  this  point  is  in 
other  ii  re-  a  line  with  the  motion  of  Venus,  and  not  in  a  line  with  the 
tard.d.  motion  Of  the  earth. 

When  the  After  Venus  passes  conjunction,  that  is,  passes  the  varying 
line  SJZ,  her  motion  becomes  retarded,  and  the  earth's  is  ac- 
celerated; but  every  motion  of  the  earth  we  ascribe  to  the  sun; 
and  in  all  modern  solar  tables,  the  corrections  of  the  sun's 
longitude  corresponding  to  the  action  of  Venus,  Mars,  Ju- 
"  e  moon,  &c.,  are  simply  the  effect  that  these  bodies 


t  by  « 
lar  pertu.-ba.  have  on  the  motion  of  the  earth. 

The  direct  effect  of  any  of  these  bodies  on  the  position  of 
the  sun  is  absolutely  insensible. 

The  relative  disturbances  of  two  planets  are  reciprocal  to 
their  masses  ;  for  if  one  is  double  in  mass  of  another,  the 


PLANETARY    PERTURBATIONS.  217 

greater  mass  will  move  but  half  as  far  as  the  smaller,  under  CHAP.  vi. 
their  mutual  action.     But  when  the  amount  of  disturbance  is 

.i  .  -i  i          Angular  i»- 

referred  to  angular  motion  for  its  measure,  regard  must  be  regularities 
had  to  the  distances  of  each  planet  from  the  sun ;  for  the  indicate  tb« 
same  distance  on  a  larger  orbit  corresponds  to  a  less  angle.*  pI^Ly  ° 
Also,  the  whole  amount  of  the  disturbing  force  of  a  superior  disturbance 
planet   on   an   inferior  will,   at  times,  be  a  tangental  force  ^ductum*"0 
(  Fig.  23  ) ;  but  the  reaction  of  the  inferior  planet  on  the  su- 
perior can  never  be  in  a  tangent  directly  with,  or  opposed  to, 
the  motion  of  the  superior. 

If  observations  can  give  the  mutual  disturbance  of  any  two 
planets,  then  these  circumstances  being  taken  into  considera- 
tion, an  easy  computation  will  give  the  relative  masses  of  the 
planets. 

( 193.)  As  a  general  result,  the  attraction  of  a  superior    The  gene- 
planet  on  an  inferior,  is  to  increase  the  time  of  revolution  of  ral  resulu  '• 

respect  to  the 

the  inferior,  and  to  maintain  it  at  a  greater  distance  from  the  tjmeg  Of  rev. 
sun  than  it  would  otherwise  have.     The  action  of  the  inferior  ointiou. 
is  to  diminish  the  time  of  revolution  of  the  superior;  and 
the  general  effect  is  greater  than  it  would  be,  if  the  inferior 
planet  were  constantly  situated  at  the  distance  of  the  sun. 
(Art.  185.) 

As  an  illustration  of  this  truth,  we  say,  that  if  Venus  were 
annihilated,  the  length  of  our  year,  and  the  times  of  revolu- 
tion of  all  its  superior  planets,  would  be  a  little  increased,  and 
the  revolution  of  Mercury,  its  inferior  planet,  would  be  a  lit- 
tle diminished.  If  Jupiter  were  annihilated,  the  times  of  re- 
volution of  all  its  inferior  planets  would  be  a  little  diminished ; 
for  it  acts  as  a  radial  force  to  keep  them  all  a  little  farther 
from  the  sun. 

( 194.)  If  the  orbits  of  all  the  planets  were  circular,  the   inequalities 
acceleration  in  one  part  of  an  orbit  would  be  exactly  compen-  m 


orbits. 


*  Geometry  demonstrates,  that,  on  the  average  of  each  revolution, 
the  proportion  in  which  this  reaction  will  affect  the  longitudes  of  the 
two  planets,  is  that  of  their  masses  multiplied  by  the  square  roots  of 
the  major  axes  of  their  orbits,  inversely;  and  this  result  of  a  very  in- 
tricate and  curious  calculation  is  fully  confirmed  by  observation.-- 
HERSOHEL. 


518  ASTRONOMY. 

C»AP.  vj.  sated  by  the  retardation  in  another ;  and  in  the  course  of  a 
whole  revolution,  the  mean  motions  of  both  planets  (the  dis- 
turber and  the  disturbed)  would  be  restored,  and  the  errors 
in  longitude  would  destroy  each  other.     But  the  orbits  are 
not  circles,  and  it  is  only  in  certain  very  rare  occurrences 
that  symmetry  on  each  side  of  the  line  of  conjunctions  takes 
place ;  and  hence,  in  a  single  revolution  the  acceleration  of 
•ds  of  Pine-  one  Par^  cannot  be  exactly  counterbalanced  by  the  retarda- 
quaiities  de-  tion  of  the  other;  and,  therefore,  there  is  commonly  left  a  cer- 
unctions  ta*a  outstanding  error,  which  increases  during  every  synodi- 
in  the  same  cal  revolution  of  the  two  planets,  until  the  conjunctions  take 
orbits       be  P^ace  iQ  °PPosite  parts  of  the  orbits,  then  it  attains  its  maxi- 
mum, which  is  as  gradually  frittered  away  as  the  line  of  con- 
junctions works  round  to  the  same  point  as  at  first. 
Some    of      Hence,  between  every  two  disturbing  planets  there  is  a  common 
uaiities  too  wwquality  depending  on  their  mutual  conjunctions,  in  the  same, 
minute  to  be  or  nearly  in  the  same,  parts  of  their  orbits.     But  it  would  be 
lce  *        folly  to  compute  the  inequalities  for  every  two  planets,  by  rea- 
son of  the  extreme  minuteness  of  the  amounts ;  for  instance, 
Mercury  is  not  sensibly  disturbed  by  Saturn  or  Uranus;  and 
Mars,  and  Mercury,  and  Uranus,  practically  speaking,  do  not 
disturb  each  other ;  but  Jupiter  and  Saturn  have  very  con- 
siderable mutual  perturbations,  on  account  of  their  orbits  be- 
ing near  each  other,  and  both  bodies  far  away  from  the  sun. 
The  effect      (195)  Again,  if  the  revolutions  of  two  planets  are  ex- 
surate^revoi  acfcly  commensurate  with  each  other,  or,  what  is  the  same 
lotions  of  the  thing,  the  mean  motion  of  both  exactly  commensurate  with 
pianeu.        ^  cjrcje>  then  the  conjunctions  of  those  two  planets  will  al- 
ways occur  at  the  same  points  of  the  orbits  (  just  as  the  con- 
junctions of  the  two  hands  of  a  clock  always  occur  at  the 
same  points  on  the  dial  plate),  and,  in  that  case,  the  conjunc- 
tions will  not  revolve  and  distribute  themselves  around  the 
orbits,  so  that  in  time,  the  radial  and  tangental  forces  will 
have  an  opportunity  to  accelerate  on  one  side  of  the  line  of 
conjunctions    as  much    as   they  retard  on  the   other;  and, 
therefore,  a  permanent  derangement  would  then  take  place. 
Aropposed      -por  in8tance  jf  three  times  the  mean  angular  motion  of 

ease  for  illos-  ,  •         -,  i 

uation.        one  planet  were  exactly  equal  to  twice  the  mean  angular  TOO- 


PLANETARY  PERTURBATIONS.  219 

tion  of  another,  then  three  revolutions  of  the  one  would  ex-  CHAP,  v 
actly  correspond  to  two  of  the  other,  and  every  second  con- 
junction of  the  two  would  take  place  in  the  same  points  of 
the  orbits ;  and  the  orbits,  not  being  circular,  the  portions  of 
them  on  each  side  of  the  line  of  conjunctions  cannot  be  sym- 
metrical, unless  the  longer  axes  of  the  two  orbits  are  in  the 
game  line,  and  the  conjunctions  also  taking  place  on  that  line. 

Here,  then,  is  a  case  showing  that  the  disturbing  force 
may  constantly  differ  in  amount  on  each  side  of  the  line  of 
conjunctions,  and,  of  course,  could  never  compensate  each 
other,  and  a  permanent  derangement  of  these  two  planets 
would  be  the  result. 

Hence,  we  perceive,  that,  to  preserve  the  solar  system,  it    stability  of 
is  necessary  that  the  orbits  should  be  circles,  or  their  times  thesolaliyt< 

t€KU» 

of  revolution  incommensurable;  but  we  do  not  pretend  to  say 
that  the  converse  of  this  is  true :  we  do  say,  however,  that  no 
natural  cause  of  destruction  has  thus  far  been  found. 

( 196.)  The  times  of  the  planetary  revolutions  are  incom- 
mensurable; but,  nevertheless,  there  are  instances  that  ap- 
proach commensurability,  and,  in  consequence,  approach  a 
derangement  in  motion,  which,  when  followed  out,  produce 
Very  long  periods  of  inequality,  called  secular  variation.  The 
most  remarkable  of  these,  and  one  which  very  much  perplexed 
the  astronomers  of  the  last  century,  is  known  by  the  term  of 
"  the  great  inequality  "  of  Jupiter  and  Saturn. 

"  It  had  long  been  remarked  by  astronomers  that,  on  com-    Th*  gnat 
paring  together  ancient  with  modern  observations  of  Jupiter  1"ein*llt1^ 
and  Saturn,  their  mean  motions  could  not  be  uniform."     The  and  Satnm. 
period  of  Saturn  appeared  to  have  been  increased  throughout 
the  whole  of  the  seventeenth  century,  and  that   of  Jupiter 
shortened.     Saturn  was  constantly  lagging  behind  its  calcu- 
lated place,  and  Jupiter  was  as  constantly  in  advance  of  his. 
On  the  other  hand,  in  the  eighteenth  century,  a  process  pre- 
cisely the  reverse  was  going  on. 

The  amount  of  retardations  and  accelerations,  corresponding     **« 
to  one,  two,  or  three  revolutions  were  not  very  great ;  but,  as  £*|£ 
they  went  on  accumulating,  material  differences,  at  length, 
existed  between  the  observed  and  calculated  places  of  both 


220  ASTRONOMY. 

.  vi.  these  planets;  and,  as  such  differences  could  not  then  be  ac- 
counted for,  they  excited  a  high  degree  of  attention,  and 
formed  the  subject  of  prize  problems  of  several  philosophical 
societies. 

Laplace  For  a  long  time  these  astonishing  facts  baffled  every  en- 
th*  deavor  to  accounfc  f°r  them,  and  some  were  on  the  point  of 
declaring  the  doctrine  of  universal  gravity  overthrown  ;  but, 
at  length,  the  immortal  Laplace  came  forward,  and  showed 
the  cause  of  these  discrepancies  to  be  in  the  near  commensu- 
rability  of  the  mean  motions  of  Jupiter  and  Saturn  ;  which 
cause  we  now  endeavor  to  bring  to  the  mind  of  the  reader  in 
a  clear  and  emphatic  manner. 

(  197.)  The  orbits  of  both  Jupiter  and  Saturn  are  ellipti- 
cal, and  their  perihelion  points  have  different  longitudes,  and, 
therefore,  their  different  points  of  conjunction  are  at  different 
distances  from  each  other,  and  no  line  *  of  conjunction  cuts  the 
two  orbits  into  two  equal  or  symmetrical  parts  ;  hence,  the 
inequalities  of  a  single  synodical  revolution  will  not  destroy 
each  other  ;  and,  to  bring  about  an  equality  of  perturbations, 
requires  a  certain  period  or  succession  of  conjunctions,  as  we 
are  about  to  explain. 
The  revo-  Five  revolutions  of  Jupiter  require  21663  days,  and  two 

'rterTd"^  of  Saturn'  21518  davs-     So  thafc>  in  a  Period  of  two  revolu- 

urn  zompar-  tions  of  Saturn  (about  sixty  of  our  years),  after  any  conjunc- 

tion of  these  two  planets,  they  will  be  in  conjunction  again  not 

many  degrees  from  where  the  former  took  place. 

Their  syno-      To  determine  definitely  where  the  third  mean  conjunction 

dicai  revoin-  wjjj  ta^e  piace  we  compute  the  svnodical  revolution  of  these 

tion      deter- 


•tinea. 


two  planets  by  dividing  the  circumference  of  the  circle  in  sec- 
onds (1296000)  by  the  difference  of  the  mean  daily  motion 
of  the  planets  in  seconds  (178".6),t  and  the  quotient  is  7253.4 
days;  three  times  this  period  is  21760  days.  In  this  period 
Jupiter  performs  five  revolutions  and  8°  6'  over;  Saturn 
makes  two  revolutions  and  8°  6'  over ;  showing  that  the  line 

*  Line  of  conjunction,  an   imaginary   line  drawn   from   the   sun 
through  the  two  planets  when  in  conjunction. 

t  See  problem  of  the  two  couriers,  Robinson's  Algebra. 


PLANETARY  PERTURBATIONS.        221 


of  conjunction  advances  8°  6'  in  longitude  during  the  period   CHA?.  TI 
of  21760  days. 

In  the  year  1800,  the  longitude  of  Jupiter's  perihelion  point 
was  11°  8',  and  that  of  Saturn  89°  9';   the  inclination  of  the 
greater  axis  of  the  orbits,  therefore,  was  78°  1'. 
Fig.  42 


Let  AB  (Fig.  42)  represent  the  major  axis  of  Saturn's    The  se 
orbit,  and  ab  that  of  Jupiter;  the  two  are  placed  at  an  angle  ^J^^""8 

Of  78°.*  plained. 

Suppose  any  conjunction  to  take  place  in  any  part  of  the 
orbits,  as  at  JS  (the  line  JS  we  call  the  line  of  conjune-  Lineofcon- 
tion) ;  in  7253.4  days  afterward  another  conjunction  will  take  Junctlon  «•• 
place.  In  this  interval,  however,  Saturn  will  describe  about 
243°  in  its  orbit,  at  a  mean  rate,  and  Jupiter  will  describe  one 
revolution  and  about  243°  over,  and  it  will  take  place  as  re- 
presented in  the  figure,  at  P  Q  (  STB  being  the  direction  of 
the  motion).  The  next  conjunction  will  be  243°  from  PQ,  or 
at  R  T.  From  RT  the  next  conjunction  will  be  at  si,  8° 6' 
in  advance  of  JS,  and  thus  the  conjunction  JS  ( so  to  speak) 
will  gradually  advance  along  on  the  orbit  from  S  to  T. 

But,  as  we  perceive,  by  inspecting  the  figure,  there  is  a 

*  We  have  very  much  exaggerated  the  eccentricities  of  these  ellip- 
ses, for  the  purpose  of  magnifying  the  principle  under  consideration. 


122 


ASTRONOMY. 


CHAP.  VI. 


The  period  of 
this  remark- 
able ine- 
quality com- 
puted, and 
the  computa- 
tion confirm- 
ed by  obser- 
vation. 


certain  portion  of  the  orbits,  between  S  and  T,  where  the  two 
planets  would  come  nearer  together  in  their  conjunction,  than 
they  do  at  conjunctions  generally,  and,  of  course,  while  any 
one  of  the  three  conjunctions  is  passing  through  that  portion 
of  the  orbits  —  Jupiter  disturbs  Saturn,  and  Saturn  reacts  on 
Jupiter  more  powerfully  than  at  other  conjunctions ;  and  this 
is  the  cause  of  "  the  great  inequality  of  Jupiter  and  Saturn" 

(  198. )  To  obtain  the  period  of  this  inequality,  we  com- 
pute the  time  requisite  for  one  of  these  lines  of  conjunction 
to  make  a  third  of  a  revolution,  that  is,  divide  120°  by  8°  6', 
and  we  shall  find  a  quotient  of  14f  £,  showing  the  period  to  be 
14f£  times  21760  days,  or  nearly  883  years:  which  would  be 
the  actual  period,  provided  the  elements  of  the  orbits  re- 
mained unchanged  during  that  time.  But  in  so  long  a  period 
the  relative  position  of  the  perigee  points  will  undergo  con- 
siderable variation ;  which  causes  the  period  to  lengthen  to 
about  918  years. 

The  maximum  amount 
of  this  inequality,  for 
the  longitude  of  Saturn, 
is  49',  and  for  Jupiter 
21',  always  opposite  in 
effect,  on  the  principle  of 
action  and  reaction. 

(199.)  The  last  gre^t 
achievement  of  the  pow- 
ers of  mind  in  the  solar 
system,  was  the  discovery 
of  the  new  planet  Nep- 
tune, by  Leverrier  and 
Adams  analyzing  the  in- 
equalities of  the  motion 
of  Uranus.  To  give1  a  rude  explanation  of  the  possibility  of 
this  problem,  we  present  Figure  43.  Let  S  be  the  sun,  and 
the  regular  curve  the  orbit  of  Uranus,  as  corresponding  to  all 
known  perturbations;  but  at  a  it  departs  from  its  computed 
track  and  runs  out  in  the  protuberance  a  c  b.  This  indicated 
that  some  attracting  body  must  be  somewhere  in  the  direction 


ABERRATION  223 

S  P,  although  no  such  body  was  ever  seen  or  known  to  exist.   CHAP>  **• 
The  next  time  the  planet  comes  round  into  the  same  portions 
D£  its  orbit,*  suppose  the  center  of  the  protuberance  to  have 
changed  to  the  line  S  0.     This  would  indicate  that  the  un-     flow  com' 

nutations 

known  arid  unseen  body  was  now  in  the  line  S  Q,  and  that  couid     be 
since  the  former  observations  it  had  changed  positions  by  the  made  for  tho 
angle  P  S  Q\  and,  by  this  angle,  and  the  time  of  its  descrip-  an     unsaen 
tion,  something  like  a  guess  could  be  made  of  the  time  of  its  planet, 
revolution. 

With  the  approximate  time  of  revolution,  and  the  help  of 
Kepler's  third  law,  its  corresponding  distance  from  the  sun 
can  be  known.  With  the  distance  of  the  unseen  body,  and 
the  amount  that  Uranus  is  drawn  from  its  orbit  by  it,  we  can 
approximate  to  its  mass. 

Thus,  we  perceive,  that  it  is  possible  to  know  much  about 
an  existing  planet,  although  so  distant  as  never  to  be  seen. 
But  the  body  that  disturbed  the  motion  of  Uranus  has  been 
teen,  and  is  called  Neptune. 


CHAPTER  VII. 

ABERRATION,    NUTATION,   AND    PRECESSION    OF   THE   EQUINOXES. 

(200.)  ABOUT  the  year  1725  Dr.  Bradley,  of  the  Green-   c±l! 
wich  observatory,  commenced  a  very  rigid  course  of  observa-  le  ,s  r' 
tions  on  the  fixed  stars,  with  the  hope  of  detecting  their  vations     on 
parallax.     These  observations  disclosed  the  fact,  that  all  the  the  fixed 

stars   for  the 

stars  which  come  to  the  upper  meridian  near  midnight,  have  purpose  of 
an  increase  of  longitude  of  about  20";  while  those  opposite,  fin(iins 
near  the  meridian  of  the  sun,  have  a  decrease  of  longitude  of  P 
20"  ;    thus  making  an  annual  displacement  of  40".     These  results. 
observations  were  continued  for  several  years,  and  found  to 
be  the  same  at  the  same  time  each  year  ;  and,  what  was  most 

•Leverrier  and  Adams  had  not  the  advantage  of  a  complete  revolu- 
tion of  Unm  UP 


tneu 


224 


ASTRONOMY. 


Aberration 
HliitraUd. 


CHAP  vn.  perplexing,  the  results  were  directly  opposite  from  such  ai 
would  arise  from  parallax. 

These  facts  were  thrown  to  the  world  as  a  problem  demand- 
ing solution,  and,  for  some  time,  it  baffled  all  attempts  at  ex- 
planation; but  it  finally  occurred  to  the  mind  of  the  Doctor, 
that  it  might  be  an  eifect  produced  by  the  progressive  motion 
of  light  combined  with  the  motion  of  the  earth ;  and,  on  strict 
examination,  this  was  found  to  be  a  satisfactory  solution. 

Fig.  44.  (201.)  A  person  stand- 

ing still  in  a  rain  shower, 
when  the  rain  falls  perpen- 
dicularly, the  drops  will 
strike  directly  on  the  top 
of  his  head;  but  if  he 
starts  and  runs  in  any  di- 
rection, the  drops  will  strike 
him  in  the  face ;  and  the 
effect  would  be  the  same, 
in  relation  to  the  direction 
of  the  drops,  as  if  the  per- 
son stood  still  and  the  rain 
came  inclined  from  the  di- 
rection he  ran. 

This  is  a  full  illustration 
of  the  principle  of  these 
changes  in  the  positions 
of  the  stars,  which  is  called 
aberration;  but  the  follow- 
ing explanation  is  more 
appropriate. 

Conceive  the  rays  of 
light  to  be  of  a  material 
substance,  and  its  particles 
progressive,  passing  from 

the  star  S  (Fig.  44)  to  the  earth  at  B-,  passing  directly 
through  the  telescope,  while  the  telescope  itself  moves  from 
A  to  B  by  the  motion  of  the  earth.  And  if  D  B  is  the  mo- 
tion of  light,  and  A  B  the  motion  of  the  earth,  then  the  tele- 


A  tother  and 
T>r«  appro* 
pi;  ate  illu»- 
3-ation. 


ABERRATION. 

icope  must  be  inclined  in  the  direction  of  A  D,  to  receive  the   CH*P-  yU- 
light  of  the  star,  and  the  apparent  place  of  the  star  would  bo 
at  &',  and  its  true  place  at  S  and  the  angled/)/? is  120". 30,  at 
its  maximum,  called  the  angle  of  aberration. 

By  the  known  motion  of  the  earth  in  its  vrbit,  we  have  tl.e 
value  of  AB  corresponding  to  one  second  of  time:  we  have 
the  angle  A  D  B  by  observation  :  the  angle  at  H  is  a  right 
angle,  arid  (  from  these  data  )  computing  the  side  B 1)  we 
have  the  velocity  of  light,  corresponding  to  one  second  of 
time.  To  make  the  computation,  we  have 

D  B  :  BA  :  :  Rad.  :  tan.  '20". 30.* 

But  B  A,  the  distance  which  the  earth  moves  in  its  orbit     The  veto 
Fig.  45.  c'ly  °'  ''s'lt 

.computed  bj 


*To  obtnin  the  lojrnrit.hmpf.ic  tanjrpnt  of  20''.?K  sw  note  on  pngo!28. 
10 


226  ASTRONOMY. 

PHAP,  vii.  in  one  second  of  time,  is  within  a  very  small  fraction  of  19 
miles;  the  logarithm  of  the  distance  is  1.278802,  and,  from 
this,  we  find  that  ED  must  be  192600  miles,  the  velocity  of 
light  in  a  second ;  a  result  very  nearly  the  same  as  before 
deduced  from  observations  on  the  eclipses  of  Jupiter's  moons. 
(Art.  143.) 

The  agreement  of  these  two  methods,  so  disconnected  and 
so  widely  different,  in  disclosing  such  a  far-hidden  and  re- 
markable truth,  is  a  striking  illustration  of  the  power  of 
science,  and  the  order,  harmony,  and  sublimity  that  pervades 
the  universe. 

A  compre.      To  show  the    effects  of  aberration  on   the  whole   starry 
hr^Ve^lCW  heavens,    we   give   figure   45.       Conceive    the   earth    to   be 

ot  tnt?  enecti 

of    aberra-  moving  in  its  orbit  from  A  to  B.     The  stars  in  the  line  AB, 
tion  whether  at  0  or  180,  are  not  affected  by  aberration.     The 

stars,  at  right  angles  to  the  line  A  JS,  are  most  affected  by 
aberration,  and  it  is  obvious  that  the  general  effect  of  aberra- 
tion is  to  give  the  stars  an  apparent  inclination  to  that  part 
of  the  heavens,  toward  which  the  earth  is  moving.  Thus 
the  star  at  90  has  its  longitude  increased,  and  the  star  op- 
posite to  it,  at  270,  has  its  longitude  decreased,  by  the  effect 
of  aberration;  both  being  thrown  more  toward  180  The  ef- 
fect on  each  star  is  20".36.  But  when  the  earth  is  in  the 
opposite  part  of  its  orbit,  and  moving  the  other  way,  from  C 
to  D,  then  the  star  at  90  is  apparently  thrown  nearer  to  0 ; 
so  also  is  the  star  at  270,  and  the  whole  annual  variation 
of  each  star,  in  respect  to  longitude,  is  40".72. 
Proofed  /  202,  )  The  supposition  of  the  earth's  annual  motion  fully 

annual     no-         \    .  ,         t  .         ,  ,  .      .      ' 

.IOH  of  ,-e  explains  aberration;  conversely,  then,  the  observed  variations 
,«..h.  of  the  stars,  called  aberration,  are  decided  proof s  of  the  earth's 

annual  motion. 

In  consequence  of  aberration,  each  star  appears  to  describe 
a  small  ellipse  in  the  heavens,  whose  semi-major  axis  is  20".36, 
and  semi-minor  axis  is  20".36  multiplied  by  the  sine  of  the 
latitude  of  the  star.  The  true  place  of  the  star  is  the  center 
of  the  ellipse.  If  the  star  is  on  the  ecliptic,  the  ellipse,  just 
mentioned,  becomes  a  straight  line  of  40".72  in  length 

If  the  star  is  at  either  pole  of  the  ecliptio,  the  ellipse  be- 


ABERRATION.  227 

comes  a  circle  of  40".72  in  diameter,  in  respect  to  a  great  CHAP,  VH 
circle ;  but  a  circle,  however  small,  around  the  pole,  will  in- 
clude all  degrees  of  longitude ;  hence  it  is  possible  for  stars 
very  near  either  pole  of  the  ecliptic,  to  change  longitude 
very  considerably,  each  year,  by  the  effect  of  aberration ;  but 
no  star  is  sufficiently  near  the  pole  to  cause  an  apparent  revo- 
lution round  the  pole  by  aberration ;  and  the  same  is  true  in 
relation  to  the  pole  of  the  celestial  equator. 

All  tftese  ellipses  have  their  longer  axes  parallel  to  the  ecliptic, 
and  for  this  reason  it  is  easy  to  compute  the  aberration  of  a 
star  in  latitude  and  longitude,*  but  it  is  a  far  more  complex 
problem  to  compute  the  effects  in  respect  to  right  ascension 
and  declination. 

(  203. )  The  aberration  of  the  sun  varies  but  a  very  little,     Aberration 
because  the  distance  to  the  sun  varies  but  little,  and  without  ° 
material  enror,  it  may  be  always  taken  at  20".*2,  subtractive. 
The  apparent  place  of  the  sun  is  always  behind  its  true  place 
by  the  whole  amount  of  aberration ;  but  the  solar  tables  give 
its  apparent  place,  which  is  the  position  generally  wanted. 

In  computing  the  effect  of  aberration  on  a  planet,  regard 
must  be  had  to  the  apparent  motion  of  the  planet  while  light 
is  passing  from  it  to  the  earth. 

The  effects  of  aberration  on  the  moon  are  too  small  to  be     The  moon 
noticed,  as  light  passes  that  distance  in  about  one  second  of  not  affected 

. .  by       aberra- 

timC-  lion. 

(  204. )  While  Dr.  Bradley  was  continuing  his  observa-     other  ine- 
tions  to  verify  his  theory  of  aberration,  he  observed  other  qnalltl"  °b- 

serveil  by  Dr. 

small  variations,  in  the  latitudes  and  declinations  of  the  stars, 
that  could  not  be  accounted  for  on  the  principle  of  ab- 
erration. 

The  period  of  these  variations  was  observed  to  be  about 


*Aber.  m  Lon. 


cos./ 

Aber.  in  Lat.  =  20".36sin.  (S—s)  sin.  /. 
In  these  expressions  S  represents  the  longitude  of  the  sun, 


9  the  longitude  of  the  star,  and  /  its  latitude. 


ASTRONOMY. 

CHAP.  m.  the  same  as  the  revolution  of  the  moon's  node,  and  the 
amount  of  the  variation  corresponded  with  particular  situa- 
tions of  the  node ;  and,  in  short,  it  was  soon  discovered  that 
the  cause  of  these  variations  was  a  slight  vibration  in  the 
earth's  axis,  caused  by  the  action  and  reaction  of  the  sun  and 
moon  on  the  protuberant  mass  of  matter  about  the  equa- 
tor, which  gives  the  earth  its  spheroidal  form,  and  the  effect 
itself  is  called  NUTATION. 

Fig.  46. 


/  205.)  We  have  shown,  in  Art.  176.  that  the  attraction 

folly  explain*      •         «     j 

ed by  the  the.  °*  a  body,  m,  on  a  ring  of  matter  around  a  sphere,  has  the 
ory  of  gravi-  effect  of  making  the  plane  of  the  ring  incline  toward  the  at- 
tracting body. 

Let  B  C,  Fig.  46,  represent  the  plane  of  the-  ecliptic  ;  and 
conceive  the  protuberant  mass  of  matter,  around  the  equator, 
tc  be  represented  by  a  ring,  as  in  the  figure.  Let  m  be  thi 


NUTATION.  229 

moon  at  its  greatest  declination,  and,  of  course,  without  the  CHAP.  VH. 
plane  of  the  ring. 

Let  P  be  the  polar  star.  The  attraction  of  m  on  the  ring 
inclines  it  to  the  moon,  and  causes  it  to  have  a  slight  motion 
on  its  center  ;  but  the  motion  of  this  ring  is  the  motion  of  the 
whole  earth,  which  must  cause  the  earth's  axis  to  change  its 
position  in  relation  to  the  star  P,  and  in  relation  to  all  the 
stars. 

When  the  moon  is  on  the  other  side  of  the  ring,  that  is, 
opposite  in  declination,  the  effect  is  to  incline  the  equator  to 
the  opposite  direction,  which  must  be,  and  is,  indicated  by  an 
apparent  motion  of  all  the  stars. 

A  slight  alternate  motion  of  all  the  stars  in  declination,  cor- 
responding to  the  declinations  of  the  sun  and  moon,  was  care- 
fully noted  by  Dr.  Bradley,  and  since  his  time  has  been  fully 
verified  and  definitely  settled  :  this  vibratory  motion  is 
known  by  the  name  of  nutation,  and  it  is  fully  and  satisfac- 
torily explained  on  the  principles  of  universal  gravity  ;  and 
conversely,  these  minute  and  delicate  facts,  so  accurately  and 
completely  conforming  to  the  theory  of  gravity,  served  as  one 
of  the  many  strong  points  of  evidence  to  establish  the  truth 
of  that  theory. 

(  206.)  By  inspecting  Fig.  46,  it  will  be  perceived  that    The  %*** 
when  the  sun  and  moon  have  their  greatest  northern  declina-  ™tation    a. 


tions,  all  the  stars  north  of  the  equator  and  in  the  same  hemi-  lustrated  by 
sphere  as  these  bodies,  will  incline  toward  the  equator;  or  all  Fls-46- 
the  stars  in  that  hemisphere  will  incline  southward,  and  those 
in  the  opposite  hemisphere  will  incline  northward  ;  the  amount 
of  vibration  of  the  axis  of  the  earth  is  only  9".6  (  as  is  shown 
by  the  motion  of  the  stars  ),  and  its  period  is  18.6,  or  about 
nineteen  years  ,  the  time  corresponding  to  the  revolution  of 
the  moon's  node.  When  the  moon  is  in  the  plane  of  the 
equator,  its  attraction  can  have  no  influence  in  changing  the 
position  of  that  plane;  and  it  is  evident  that  the  greatest  ef- 
feet  must  be  when  the  declination  is  greatest.  node  most  b« 

The  moon's  declination  is  greatest  when  the  longitude  of  tocorrespond 

.  •  /•    A    •         tothemoon'* 

the  moon  s  ascending  node  is  0,  or  at  the  first  point  of  Aries. 
The  greatest  declination  is  then  28°  on  each  side  of  the 


23o  ASTRONOMY. 

CHAP.  vn.  equator;  but  when  the  descending  node  is  in  the  same  point, 
the  moon's  greatest  declination  is  only  18°.  Hence  there  will 
be  times,  a  succession  of  years,  when  the  moon's  action  on  the 
protuberant  matter  about  the  equator  must  be  greater  than  in 
an  opposite  succession  of  years,  when  the  node  is  in  an  oppo- 
site position.  Hence,  the  amount  of  lunar  nutation  depends 
on  the  position  of  the  moon's  nodes. 

Monthly  nn-      j^  js  very  natural  to  suppose  that  the  period  of  lunar  nuta- 

gmaii/  tion  would  be  simply  the  time  of  the  revolution  of  the  moon ; 
and  to,  in  fact,  it  is ;  but  the  corresponding  amount  is  very 
small  only  about  one-tenth  of  a  second.  This  is  because  half 
a  lunar  revolution,  about  13i  days,  while  the  moon  is  on  one 
side  of  the  equator,  is  not  a  sufficient  length  of  time  for  the 
moon  to  effect  much  more  than  to  overcome  the  inertia  of  the 
earth ;  but,  in  the  space  of  nine  years,  effecting  a  little  more 
than  a  mean  result  at  every  revolution,  the  amount  can  rise  to 
9". 6,  a  perceptible  and  measurable  quantity. 
The  mean  (  207.)  The  mean  course  of  the  moon  is  along  the  ecliptic : 

effect  of  the  jts  variation  from  that  line  is  only  about  five  degrees  on  each 

orioon  on  the 

mass  of  mat.  side  \  hence,  the  medium  effect  of  the  moon  on  the  protuberant 
ter  around  mass  of  matter  at  the  equator  is  the  same  as  though  the 
moon  was  all  the  while  in  the  ecliptic.  But,  in  that  case,  its 
effect  would  be  the  same  at  every  revolution  of  the  moon ; 
and  the  earth's  equator  and  axis  would  then  have  an  equili- 
brium ofpos:tion,  and  there  would  be  no  nutation,  save  the 
slight  monthly  nutation  just  mentioned,  which  is  too  small  to 
be  sensible  to  observation ;  and  the  nutation  that  we  observe, 
is  only  an  inequality  of  the  moon's  attraction  on  the  protube- 
rant equatorial  ring ;  and,  however  great  that  attraction  might 
be,  it  would  cause  no  vibration  in  the  position  of  the  earth, 
if  it  were  constantly  the  same. 

Solar  nn.  We  have,  thus  far,  made  particular  mention  of  the  moon, 
tolion  but  there  is  also  a  solar  nutation :  its  period  is,  of  course,  a 
year ;  and  it  is  very  trifling  in  amount,  because  the  sun  at 
tracts  all  parts  of  the  earth  nearly  alike;  and  the  short 
period  of  one  year,  or  half  a  year  (which  is  the  time  that  the 
unequal  attraction  tends  to  change  the  plane  of  the  ring  iu 


THE  EQUINOXES.  231 

one  direction),  is  too  short  a  time  to  have  any  great  effect  on  CHAP.  vil. 
the  inertia  of  the  earth. 

The  solar  nutation,  in  respect  to  declination,  is  only  one 
second. 

(208.)  Hitherto  we  have  considered  only  one  effect  of  nu- 
tation —  that  which  changes  the  position  of  the  plane  of  the 
equator  —  or,  what  is  the  same  thing,  that  which  changes  the 
position  of  the  earth's  axis  ;  but  there  is  another  effect,  of 
greater  magnitude,  earlier  discovered,  and  better  known,  re- 
sulting from  the  same  physical  cause,  we  mean  the 

PRECESSION    OF   THE   EQUINOXES. 

We  again  return  to  first  principles,  and  consider  the  mu-     FiMt  pna- 
tual  attraction  between  a  ring  of  matter  and  a  body  situated  CIJ 
out  of  the  plane  of  the  ring;  the  effect,  as  we  have  several 
times  shown,  is  to  incline  the  ring  to  the  body,  or  (which  is 
the  same  in  respect  to  relative  positions),  the  body  inclines 
to  run  to  the  plane  of  the  ring. 

The  mean  uttraction  of  the  moon  is  m  the  plane  of  the     The  mean 
ecliptic.     The  sun  is  all  the  while  in  the  ecliptic.     Hence,  the 


„» 


mean  attraction  of  both  sun  and  moon  is  in  one  plane,  the  moon  are  in 
ecliptic;  but  the  equator,  considered  as  a  ring  of  matter  sur- 
rounding  a  sphere,  is  inclined  to  the  plane  of  the  ecliptic  by 
an  angle  of  23£  degrees,  and  hence  the  sun  and  moon  have  a 
constant  tendency  to  draw  the  equator  to  the  ecliptic,  and 
actually  do  draw  it  to  that  plane  ;  and  the  visible  effect  is, 
to  make  both  sun  and  moon,  in  revolutions,  cross  the  equator 
sooner  than  they  otherwise  would,  and  thus  the  equinox  falls 
back  on  the  ecliptic,  called  the  precession  of  the  equinoxes. 

The  annual  mean  precession  of  the  equinoxes  is  50".  1  of    The  Precc«- 
,          .      Ai  •  ,     .1  •  »ion  of  *• 

arc,  as  is  shown  by  the    sun  coming  into  the    equinox,  or  equinoxet 

crossing  the  equator  at  a  point  50".  1   before  it  makes  a  revo- 
lution in  respect  to  the  stars. 

Perhaps  it  is  clearer  to  the  mind  to  say,  that  the  sun  is     Natural 
drawn  to   the  equator  by  the  protuberant  mass  of  matter  pregticil> 
around  the  earth,  and,  in  consequence,  arrives  at  the  equator, 
in  its  apparent  revolutions,  sooner  than  it  otherwise  would. 
But  the  truth  is,  that  the  ecliptic  is  stationary  in  position, 


232  A  S  T  R  0  N  O  M  Y 

CHAP,  vii.  and  the  equator,  by  a  slight  motion, meets  the  ecliptic;  which 
motion  is  caused  by  the  attractions  of  the  sun  and  moon,  as 
has  been  several  times  explained, 
rhe  tine       j^  t^e  1110on  were  all  the  while  in  the  ecliptic,  the  preees- 

physical 

cause  of  the  sion  of  the  equinoxes  would  then  be  a  constantly  flowing  quan- 
,*ece»«,ion  of  ^     ec    aj  to  5Q//^    for  each  year :  but,  for  a  succession  of 

the  equinox- 

es  about  nine  years,  the  moon  runs  out  to  a  greater  declination 

than  the  ecliptic,  and,  during  that  time,  its  action  on  the 
equatorial  matter  is  greater  than  the  mean  action,  and  then 
comes  a  succession  of  about  nine  years,  when  its  action  is 
less  than  its  mean ;  hence,  for  nine  years,  the  precession  of 
the  equinoxes  will  be  more  than  50". 1  per  year,  and,  for  the 
nine  years  following,  the  precession  will  be  less  than  50".  1 
for  each  year ;  and  the  whole  amount  of  variation,  for  tJiis  in- 
equality, in  respect  to  longitude,  is  17". 3,  and  its  period  is  half 
a  revolution  of  the  moon's  nodes.  This  inequality  is  called 
the  equation  of  the  equinoxes,  and  varies  as  the  sine  of  the 
longitude  of  the  moon's  nodes. 

Equation        The  equation  of  the  equinoxes,  of  course,  affects  the  length 
equl"  of  the  tropical  year,  and  slightly,  very  slightly,  affects  side- 
real time. 

Mean  and      There  is  a  true  equinox  ajid  a  mean  equinox;  and,  as  side- 
8  real  time  is  measured  from  the  meridian  transit  of  the  equi- 
nox, there  must  be  a  true  sidereal  and  a  mean  sidereal  time; 
but  the  difference  is  never  more  than  1.1  s.  in  time,  and,  gene- 
rally, it  is  much  less. 
Explanation      (  209.)  In  the  hope  of  being  more  clear  than  some  authors 

rFlg'47  have  been,  in  explaining  the  results  of  precession,  we  present 
Fig.  47.  E  represents  the  pole  of  the  ecliptic,  and  the  great 
circle  around  it  is  the  ecliptic  itself.  P  is  the  pole  of  the 
earth,  23°  27'  from  the  pole  E,  and  around  P,  as  a  center,  we 
have  attempted  to  represent  the  .equator,  but  this,  of  course, 
is  a  little  distorted;  qp  and  ^  are  the  two  opposite  points 
where  the  ecliptic  and  equator  intersect;  <yE  is  the  first  me- 
ridian of  longitude;  <pP  is  the  first  meridian  of  right  ascen- 
sion. The  angle  JE^pP  is  23°  27',  and  E  P,  produced,  is  the 
meridian  passing  through  the  solstitial  points.  To  obtain  a 
clear  conception  of  the  precession  of  the  equinoxes,  the  stars 


THE    EQUINOXES. 


233 


the  ecliptic,  and   its  pole  E,  must  be  considered  as  FIXED,  CHAI>.  vii 
and  the  line  qp  =£=  as  having  a  slow  motion  of  50". 1   per  an- 

Fig.  47. 


From     the 

num,  on  the  ecliptic,  in  a  retrograde  direction ;  and  this  must  fixcd  pogi. 
carry  the  pole  P,  around  the  point  E,  as  a  center,  carrying  tion  of  th« 
also  the  solstitial  points  backward  on  the  ecliptic.  Some  ^S'0P  '^'f  ^e 
of  the  stars  have  proper  motions;  but,  putting  that  circum-  stars,  the 
stance  out  of  the  question,  the  stars  are  fixed,  and  the  eclip-  st 
tic  is  fixed;  therefore,  the  stars  never  change  latitude,  but  tnde 


234  ASTRONOMY. 


.^YII.  the  whole  frame-work  of  meridians  from  the  pole  P,  the  pole 
itself,  and  the  equator,  revolve  over  the  stars  ;  and,  in  respect 
to  that  motion  of  the  meridian  and  the  equator,  the  stars 
change  right  ascension,  declination,  and  longitude,  but  do  not 
change  latitude.     The  stars  change  longitude,  simply  because 
the  first  meridian  of  longitude,  T  E,  moves  backward  ;  they 
change  right  ascension,  because  the  meridian,  cp  P,  and  all 
the  meridians  of  right  ascension,  revolve  backward. 
One  hemi.      By  inspecting  the  figure,  we  readily  perceive  that  all  tbe 
•un1"     ap-  stars  near  V  must,  apparently,  approach  the  north  pole,  be- 
proaches  the  cause  the  pole,  in  its  revolution  round  E,  is  approaching  to- 
thertother°re'  war(^  ^at  Par^  °^  *he  ecliptic  ;  for  the  same  reason,  all  the 
cede*    from  stars  near  ^  are,  apparently,  moving  southward,  because  the 
equator  is  being  drawn  over  them.     In  short,  all  the  stars, 
from  the  eighteenth  hour  of  right  ascension  through  <Y>,  to 
the  sixth  hour  of  right  ascension,  must  diminish  in  north  po- 
lar distance,  and  all  the  stars,  from  the  six  hours  through  *±  , 
to  the  eighteenth  hour  of  right  ascension,  must  increase  in 
north  polar  distance,  in  consequence  of  the  precession  of  the 
equinoxes. 

inspection  These  observations  may  be  confirmed  by  inspecting  Table 
II,  in  which  is  registered  the  positions  of  the  principal  fixed 
stars,  with  their  annual  variations.  The  column  of  annual 
variation  of  declination  changes  sign  at  the  point  correspond- 
ing to  six  hours,  and  eighteen  hours  of  right  ascension  ;  and 
the  rapidity  of  this  variation  is  greater  as  the  star  is  nearer 
to  0  hours,  or  twelve  hours  of  right  ascension. 

Annual  va-  When  the  right  ascension  of  a  star  is  0  hours,  or  twelve 
hours,  it  is  easy  to  compute  its  annual  variation  in  declina- 


how  comput-  tion,  corresponding  to  its  precession  along  the  ecliptic  of 
50".l.  Conceive  a  small  plane  triangle  whose  hypothenuse  is 
50".l,  the  angle  at  the  base  23°  27'  40"  (».  e.  the  obliquity 
of  the  ecliptic  ),  the  side  opposite  to  this  angle  will  be  found 
to  be  a  little  over  20",  corresponding  to  the  figures  in  the 
table. 

Proper  mo.  jfc  jg  thus,  by  the  motion  of  these  imaginary  lines  over  the 
wn°le  concave  of  the  heavens,  that  the  annual  variation  of 
both  right  ascension  and  declination  of  each  individual  star 


THE  EQUINOXES.  235 

in  the  catalogue  is  computed  and  put  down  ;  and  if  any  par-  CHAP,  vn 
ticular  star  does  not  correspond  with  this,  it  is  said  to  have 
proper  motion  ;  and  it  is  thus  that  proper  motions  are  detected. 

As  P  must  circulate  round  E  by  the  slow  motion  of  50".  1  Final  effect 
in  a  year,  it  will  require  25868  years  to  perform  a  revolution; 
and  the  reader  can  perceive,  by  inspecting  the  figure,  why 
the  pole  star  is  in  apparent  motion  in  respect  to  the  pole,  and 
why  that  star  will  cease  to  be  the  polar  star,  and  why,  at  the 
expiration  of  about  12000  years,  the  bright  star,  Lyra,  will 
be  the  polar  star. 

(  210.)  The  mean  effect  of  the  moon  in  producing  the  pre- 


son. 


cession  of  the  equinoxes  is,  to  the  mean  effect  of  the  sun,  as  tive  effect  of 
five  to  two.     The  sun's  action  is  nearly  constant,  because 


the  sun  is  always  in  the  ecliptic ;  a  small  annual  variation, 
however,  is  observed.  The  great  inequality  of  17".3,  corre- 
sponding to  about  nineteen  years,  is  caused  entirely  by  the 
unequal  action  of  the  moon,  depending  on  the  longitude  of 
the  moon's  ascending  node. 

In  consequence  of  this  inequality,  the  pole,  P,  does  not 
move  round  the  pole  of  the  ecliptic,  E,  in  an  even  circumfe-  motion  of  th» 
rence  of  a  circle,  but  it  has  a  waving  or  undulating  motion,  as  *"nnj  "'* 
represented  in  this  figure ;  each  wave  pole  of  the 

corresponding  to  nineteen  years ;  and,  ^^ — -^^^  ecliptic, 

therefore,  there  must  be  as  many  of 
them  in  the  whole  circle  as  19  is  con- 
tained in  25868.  From  this,  we  per- 
ceive, that  the  undulations  in  the  fig- 
ure are  much  exaggerated,  and  vastly 
too  few  in  number;  an  exact  linear 
representation  of  them  would  be  im- 
possible. 

(211.)  From  the  foregoing,  we  learn  that  the  positions  of     Mean  an 
all  the  stars  are  affected  by  aberration,  precession,  and  nuta-  »pp»»nt 
tion :  the  amount  for  each  cause  is  very  trifling  in  itself,  yet,  star> 
in  most  cases,  too  great  to  be  neglected,  when  accuracy  is 
required ;  and  it  is  as  difficult  to  make  computations  for  a 
small  quantity  as  for  a  large  one,  and  often  greater;  and  to 
reduce  the  apparent  place  of  a  fixed  star  from  its  mean  place, 


236  ASTRONOMY. 

CHAP,  vn   and  its  me?*n  place  from  its  apparent  place,  is  one  of  the  most 

troublesome  problems  in  practical  astronomy. 
Genera,  for-      ^he  mean  place  of  a  fixed  star,  reduced  to  the  time  of  ob- 

enulae,  where  ..-..,  .  . 

found  servation,  is  sufficiently  near  its  apparent  place   to  be  con- 

sidered the  same.  The  practical  astronomer,  however,  who 
requires  the  star  as  a  point  of  reference,  or  uses  it  for  the 
adjustment  of  his  instruments,  must  not  omit  any  cause  of 
variation;  but  such  persons  will  always  have  the  aid  of  a 
Nautical  Almanac,  where  general  formulae  and  tables  will  be 
found,  to  direct  and  facilitate  all  the  requisite  reductions. 
Importance  (  212.)  Physical  astronomy  brings  many  things  to  light 
that  would  otherwise  escape  observation,  and  some  of  these 


developments,  at  first,  strike  the  learner  with  surprise,  and  he 
is  not  always  ready  to  yield  his  assent.  For  instance,  as  a 
general  student,  he  learns  that  the  anomalistic  year,  the  time 
that  the  earth  moves  from  its  perigee  to  its  perigee  again,  is 
365  d.  6  h.  14m.  ;  that  the  perigee  is  very  slow  in  its  motion, 
moves  only  about  12"  in  a  year,  and  is  subject  to  but  few 
fluctuations.  He  has  also  learned  that  the  earth,  in  its  orbit, 
describes  equal  areas  in  equal  times;  hence,  he  concludes, 
that  the  time  from  perigee  to  perigee,  or  from  apogee  to  apo- 
gee, must  be  very  nearly  a  constant  quantity;  but,  on  con- 
sulting and  comparing  the  predictions  to  be  found  in  the  En- 
glish nautical  almanacs,  he  will  find  these  periods  to  be  (in 
comparison  to  his  anticipations)  very  fluctuating.  They 
differ  from  the  state*  mean  times,  not  only  by  minutes  and 
seconds,  but  by  hours,  and  even  days.  The  investigator  is, 
at  first,  surprised,  and  fancies  a  mistake;  at  least,  a  mis- 
print; but,  on  examining  concurrent  facts,  such  as  the  lo- 
garithms of  the  distance  from  the  sun,  and  the  sun's  true 
motion  at  the  time,  he  finds  that,  if  a  mistake  has  been  made, 
it  is  a  very  harmonious  one,  and  every  other  circumstance  has 
been  adapted  to  it. 
The  lati.  But  let  us  turn  a  moment  from  these  facts,  and  examine 

explain.  tne  firsfc  Page  of  our  Tables-     Tnere  Jt  wil1  De  found,  that  the 

.  sun  has  latitude  ;  that  it  deviates  to  the  north  and  south  of 

the  ecliptic,  by  a  quantity  too  small  ever  to  le  observed  :  it  is, 

therefore,  a  quantity  wholly  determined  by  theory,  and,  as 


THE   EQUINOXES.  237 

the  sun's  latitude  changes  with  the  latitude  of  the  moon,  we  CHAP,  vn 
must  seek  for  its  cause  in  the  lunar  motions.  Fig.  48. 

To  understand  the  fact  of  the  sun  having 
latitude,  we  must  admit  that  it  is  the  center 
of  gravity  between  the  earth  and  moon,  that 
moves  in  an  elliptical  orbit  round  the  sun; 
and  that  center  is  always  in  the  ecliptic ;  and 
the  sun,  viewed  from  that  point,  would  have 
no  latitude.  But  when  the  moon,  m,  (  Fig. 
48  ),  is  on  one  side  of  the  plane  of  the  eclip- 
tic, Sc,  the  earth,  E  would  be  on  the  other  I 
side,  and  the  sun,  seen  from  the  center  of  the] 
earth,  would  appear  to  lie  on  the  same  side 
of  the  ecliptic  as  the  moon.  Hence,  the  sun 
will  change  his  latitude,  when  tlie  moon  changes 
her  latitude. 

If  the  moon  were  all  the  while  in  the  plane  of  the  ecliptic,      u»n?itnd» 
the  sun  would  have  no  latitude  (save  some  extremely  minute  °fthesun*f 

v  fecled  by  th« 

quantities,  from  the  action  of  the  planets,  when  not  in  the  position  of 
plane  of  the  ecliptic  )  ;  but  the  moon  does  not  deviate  more  the  moon' 
than  5°  20  from  the  ecliptic,  and,  of  course,  the  earth  makes 
but  a  proportional  deviation  on  the  other  side ;  but.  in  longi- 
tude, the  moon  deviates  to  a  right  angle  on  both  sides,  in  re- 
spect to  the  sun,  and  when  the  moon  is  in  advance  in  respect 
to  longitude,  the  sun  appears  to  be  in  advance  also ;  and 
when  the  moon  is  at  her  third  quarter,  the  longitude  of  the 
sun  is  apparently  thrown  back  by  her  influence : — the  great- 
est variation  in  the  sun's  longitude,  arising  from  the  motion 
of  the  earth  and  moon  about  their  center  of  gravity,  is  about 
6"  each  side  of  the  mean.  Now  it  is  this  motion  of  the  Loneitlul<> 

of  the  moon 

earth  around  the  common  center  of  gravity  of  the  earth  and  affects     the 
moon,  that  chiefly  affects  the  time  when  the  earth  comes  to  timo  that  the 

-,  .  TTI  i  •       «  •         earth    comei 

its  apogee  and  perigee.      >\  hen  the  moon  is  in  conjunction  to  JM  apogM 
with  the  sun,  the  center  of  the  earth  is  farther  from  the  sun  and 
than  it  otherwise  would  be ;  and  when  the  moon  is  in  oppo- 
sition to  the  sun,  the  earth  is  about  3200  miles  nearer  the 
sun  than  it  would  be  in  its  mean  orbit ;  and  thus,  we  per- 
ceive, that  the  longitude  of  the  moon  has  a  great  influence  in 


238  ASTRONOMY. 

CHAP.  vn.  bringing  the  earth  into,  or  preventing  it  from  coming  into,  it» 
perigee  or  apogee ;  but  the  perigee  and  apogee  points,/or  the 
center  of  gravity,  are  quite  uniform,  agreeably  to  the  views  ex- 
pressed in  the  first  part  of  this  article.  These  explanations 
will  give  a  general  insight  into  some  of  the  apparent  intrica- 
cies of  physical  astronomy. 

Small  equa-  The  small  equations  of  the  sun's  center  are  computed  on 
nnns  of  lie  ^e  principle  explained  by  Fig.  48,  the  sun  having  a  mo- 
tion  round  the  center  of  gravity  between  itself  and  each  of 
the  planets.  For  example,  the  perturbation  produced  by  Ju- 
piter is  greatest  when  Jupiter  is  in  longitude  90°  from  the 
sun,  as  seen  from  the  earth ;  the  greatest  effect  is  then  about 
8",  and  varies  very  nearly  as  the  sine  of  Jupiter's  elongation 
from  the  sun. 

When  Jupiter  is  in  conjunction  with  the  sun,  the  sun  is 
nearer  the  earth  than  it  otherwise  would  be;  and,  on  this  ac- 
count, we  have  a  small  table  to  correct  the  sun's  distance 
from  the  earth,  called  the  perturbations  of  the  sun's  distance. 

The  same  remarks  apply  to  other  planets  but,  to  avoid 
confusion,  the  effects  of  each  one  must  be  computed  sepa- 
rately. 


PRACTICAL   ASTRONOMY  «39 

SECTION   IV. 
PRACTICAL    ASTRONOMY. 

PREPARATORY    REMARKS. 

WE  have  now  done  with  general  demonstrations,  and  with  TKM> 
minute  and  consecutive  explanations ;  but  we  shall  give  all 
necessary  elucidation  in  relation  to  the  particular  problems 
under  consideration.  To  go  through  this  part  of  astronomy 
with  success  and  satisfaction,  the  reader  must  have  a  passa- 
ble understanding  of  plane  and  spherical  trigonometry :  and 
if  to  these  he  adds  a  general  knowledge  of  the  solar  system, 
as  taught  in  the  foregoing  pages,  he  will  have  a  full  compre- 
hension of  all  we  design  to  embrace  in  this  section. 

To  prompt  the  student  in  his  knowledge  of  trigonometry. 
w<?  give  the  following  formulae : 

I.     Relative  to  a  single  arc  or  angle. 
1.     -     -     -     sin.  a  =  tan.  a  cos.  a.* 


tan.  a 


l-|-cos.  a 
7.     -     -     -     sin.  2a= 2  sin.  a  cos.  a. 


Radius  is  unity  in  all  these  equation!. 


8.      - 


ASTRONOMY. 

cos.  2a=2  cos.  za — 1=1 — 2  sin. 


II.     Relative  to  two  arcs,  a  and  b,  of  which  a  is  supposed 
to  be  the  greater. 

9.  -  gin.  (a-j-&j=siii.  a  cos.  6-}-sin.£  oos.  a. 

10.  -  cos.  (<7-j-6)=cos.  a  cos  b — sin.  a  sin.  b. 

11.  -  sin.  (a — b/=sin.a  cos.  b — sin.  b  cos.  a. 

12.  -  cos.  (a — A)  =  cos.  a  cos.  d-f-sin.  a  sin.  b. 
Sum  of  (9  )  and  (  11  )  gives  13,  diff.  gives  14 

13.  -  sin.  (a-j-i)-f-sin.  (a — i)=2  sin.  a  cos.  b. 

14.  -  sin.  (a-f-5) — sin.  (a — i)=2  cos.  a  sin.  A. 

an.  A 


15.  -     tan 

16.  -     tan.  (a  —  6) 
_ 


sin.  <7_f_sin.  b 
sin.  a  —  sin.  b 


18.     - 


tan.  a-{-tan.  b 
tan.  a — tan-  b 


tan. 

— ^ 

1 — tan. a  tan.  bj 

tan.tf — tan.  b 
1-4-tan.  a  tan.  b' 

tan.  -i  (tf_j_A) 
tan.i  (a — b)' 

sin. 


19. 


{   1-4-t 
l=t 


1-4-tan.  £ 
— tan.  b 


\    sin.  (a — b\ 
=tan. 


1— tan. 
I 


=tan.  (45°— b). 


We  shall,  probably,  make  an  application  of  the  fo 
theorem ;  it  applies  to  finding  the  unknown  angles  of  a  tri- 
angle, when  the  log  irithms  of  two  sides  (not  the  sides  them- 
selves) and  the  angle  included  between  the  sides  are  given. 

The  greater  of  two  sides  of  a  plane  triangle  is,  to  ilie  less, 
as  radius  to  the  tangent  of  a  certain  angle.  Take  this  angle 
from  45°,  and  call  the  difference  a.  Lastly,  radius  is  to  the 
tangent,  a,  as  the  tangent  of  the  half  sum  of  the  angles  at  tin 
base  is  to  the  tangent  of  half  their  difference. 

TIT.     Resolution  of  right-angled  spherical  triangles. 

In  the  following  equations,  h  is  the  hypothenuse,  s  a  given 


EQUATIONS. 


241 


side,  a,  a  given  angle,  and  x  the  quantity  sought.     (TV  right      TWO. 
angle  is  unity,  and  always  given.) 


Given, 

Required, 

Solution. 

h 

'  side  op.  a 

20.  sin.  #=sin.  A  sin.  a. 

and 

side  adj.  a 

21.  tan.  ar=  tan.  A  cos.  r 

a 

,  the 

other  angle 

22.  cot.  2=  cos.  A  tan.  r 

the 

other  side 

23.  cos.  *=?—  h 

it 

cos.  * 

and    • 

ang 

.  adj.  * 

24.  cos.  ar=tan.  s  cot.  A 

1 

ne                sin.  * 

ang 

.  Op.  8 

25.    BID.  *=-;  ; 

sin.  A 

•          r 

sin  * 

o 

h 

26.  sin.  *=!^ll 

and 

sin.  a 

a     1 

tne 

other  side, 

27.  sin.  g=tan.  «  cot.  a 

opposite 

*hp 

n  flint*   nnrr 

28   sin  x     C°S>  a 

"r^           ' 

bile 

utncr  ciijg. 

COS.  9 

/d     ! 

h 

29.  cot.  #=cos.  a  cot.  * 

ana 

the 

other  side, 

30.  tan.  2=  tan.  a  sin.  « 

«  rl  i  a  r*fi  n  f       1 

the 

other  ang. 

31.  cos.  #=sin.  a  cos.  *. 

The       (  h 

two  sides.  (  the  angles, 


32.  cos.  #=cos.  s  cos.  other  side 

33.  cot.  ar=sin.  adj.  sideXcot. 

[opp.  side. 

IV.  Resolution  of  oblique  angled  spherical  triangles. 
Let  A  B  and  C  be  the  three  angles  of  any  spherical  triangle, 
and  a  b  and  C  the  sides  opposite  to  them  respectively,  that  is, 
the  side  a  is  opposite  to  A,  &c. 

In  spherical  trigonometry, the  sines  of  the  angles  are  propor- 
tional to  the  sines  of  the  opposite  sides. 

sin.  A sin.  B       sin.  0 

Bin.  a       sin.  b        sin.c 


_..       ..        0 .    sin.  -a       sin.  j. 
Therefore  34.  — =  - — - 


Given  the  three  sides  abc; 
Required  one  of  the  angles,  A. 

35.    -  -   Sin.  a  i  A  =  - 
1U 


sm.b  sin.  c 


242 


ASTRONOMY. 


Tuo. 

.  36.  -    - 


sin.  b  sin.  c 


In  35  and  36, 


CHAPTER    I. 


ASTRONOMICAL   PROBLEMS. 
PROBLEM  L 

CHAP.  I.  Given  the  right  ascension  and  declination  of  any  heavenly 
body, to  find  its  latitude  and  longitude  ;  or  conversely,  given  tht 
latitude  and  longitude  of  a  body  to  find  its  corresponding  righ, 
ascension  and  declination. 

Fig.  49. 

From  any  point  as  a  center 

(Fig.  49)  describe  a  circle  Q 
EPo$,  &c.  Let  this  circle 
represent  the  meridian,  which 
passes  through  the  pole  of  the 
ecliptic  E,  the  pole  of  the 
earth's  axis  P,  and  through  the 
solstitial  points  05  and  y?. 
Then  the  point  Aries  (  <Y>)  will 
be  at  the  center  of  the  circle, 
and  V?  °TP  25  and  Q  <p  q  will  be 

lines  crossing  each  other  by  an  angle  equal  to  the  obliquity 
of  the  ecliptic.  P p  is  the  celestial  meridian  which  passes 
through  the  equinoctial  points,  and  is  the  first  meridian  of 
right  ascension.  E  qp  e  is  the  first  meridian  of  longitude,  and, 
of  course,  the  angle  E  qp  P  is  equal  to  the  obliquity  of  the 
ecliptic. 

Let  s  be  the  position  of  any  celestial  body,  and  draw  the 
meridian  of  right  ascension  Psp;  also  draw  the  meridian  of 
longitude  Es  e  draw  also  «p  *.  We  have  now  two  right-angled 
spherical  triangles  *  D  T  and  qp  Bs,  having  a  common  hypo- 
thenuse  T  *;  the  first  is  the  right  ascension  triangle,  the 


The  figure 
is  consiileieil 
transparent, 
and  both 
•ides  of  it  are 
represented. 


PRACTICAL     PROBLEMS.  243 

second  is  the  longitude  triangle.  Let  the  student  observe  CHAT.L 
that  the  line  Q  g  represents  a  circle,  the  whole  equator ;  and 
the  point  <TP  represents,  in  fact,  two  points,  the  0  degree  of 
right  ascension  and  the  180th  degree.  So  the  point  s  repre- 
sents two  points,  and  T  D  is  the  right  ascension  from  0  de- 
gree, or  from  180  degrees. 

ID  our  figure,  the  point  *  is  north  of  both  ecliptic  and 
equator ;  but  it  might  have  been  between  the  two,  or  south  of 
both ;  hence,  to  meet  every  case,  the  judgment  of  the  opera- 
tor must  be  called  into  exercise  to  perceive  a  general 
solution. 

Now,  having  the  right  ascension  and  declination  of  *,  we 
find  its  latitude  and  longitude  thus : 

In  the  triangle  qp  Ds,  <v>  D  and  Ds  are  given,  and  equa- 
tion 32  gives  <TP  s  ( h ) ;  33  gives  the  angle  s  <y  •#•  From 
s<¥>  D  subtract  B  <1P  D,  the  obliquity  of  the  ecliptic,  and 
there  remains  the  angle  s  T  B.* 

With  the  angle  s  *(>  B  and  the  side  <p  s,  equation  20 
gives  sB  the  latitude,  and  21  gives  <vB  the  longitude. 

EXAMPLES  . 

1.  The  right  ascension  of  a  certain  point  in  the  heavens  is 
5  h.  7  m.  50  s.,  or  in  arc  76°  57'  30" ;  and  its  declination  is 
26°  11'  36"  N. : 

What  is  the  latitude  and  longitude  of  the  same  point  ?  Foar  e 

(32.)  (33.)       tions     COB 

rD  76°  57'  30"  cos.  9.353454      -     -    -      sin.  9.988651 
sD  26°  11'  36"  cos.  9.952952       -    -    -      cot.  10.308104 
np  s  78°  19'    3"  cos.  9.306406      26° 47'  27"cot.  10.296755 
....    23   27  32 
....       3   19  55  =  a 

*In  general,  take  the  difference  between  the  angle  8<¥>D  and  the 
obliquity  of  the  ecliptic  ;  and  if  the  angle  S  T  D  is  the  greater  quan- 
tity, the  body  is  north  of  the  ecliptic,  otherwise  it  is  south  of  it 
When  the  declination  it  south,  the  angle  S^  D  must  be  added  to  the 
obliquity  of  the  ecliptic  in  the  first  and  second  quadrants,  and  sub- 
tracted in  the  third  and  fourth.  Hence  the  judgment  of  the  operator 
uinst  be  called  in  to  decide  the  particulars  of  the  case ;  or  he  must 
have  a  general  formula  that  will  give  no  exercise  to  the  mind. 


«44  ASTRONOMY. 

CHAP   t  (20.)  (21.) 

(A)  78°  19'  3"  sin.  9.990911  tan.  10.684611 

(a)    3   19  55    sin.  8.763965  cos.   9.999265 

3°l¥W'  sin.  8.754876    78  18  6  tan.  10.683876 

Thus  we  determine  that  the  longitude  must  be  78°  18'  6", 
and  the  latitude  3°  15'  36"  N. 

2.  The  longitude  of  the  moon,  at  a  certain  time,  according  to 
computation,  VMS  102°  7';  and  latitude  5°  14'  15"  S.  : 

What  was  the  corresponding  right  ascension  and  declination  ?* 


From  the,. 


(33.) 


examples  we 

V  B  77°  53' 

cos 

.9.322019 

sin. 

9.990215 

might  form  a 
general  rale  * 

sB    5° 

14' 

15" 

COS 

.9.998183 

cot. 

11.037780 

T«77° 

56' 

12" 

COS, 

,  9.320202 

5° 

21' 

27" 

cot. 

11.027995 

form 
dom 

ed     sel- 
reflect 

BwD  -    - 

23 

27 

42 

principles  ; 

18 

6 

15 

then  " 
edoc 

sfore   for 
sational 

(20.) 

(21.) 

purposes,  we 

(A)  77°  56' 

12" 

sin. 

9.990302 

tan. 

10.670170 

the 

back  on 

nrimarv 

(a)  18 

6 

15 

sin. 

9.492400 

cos. 

9.977948 

equations. 

17 

41 

22 

sin. 

9.482702 

77°19'41" 

tan. 

10.648118 

Thus  we  find  that  the  right  ascension  distance  on  the  equa* 
tor,  from  the  180th  degree,  was  77°  19'  41";  or  its  right  as- 
cension in  arc  was  102°  40'  19",  or  in  time,  6h.  50m.  41s. 

3.  By  meridian  observations  on  the  moon,  at  a  certain  time, 
its  right  ascension  was  found  to  be  16h.  53m.  33s.,  and  its  decli- 
nation 17°  51'  36".  S.  :  what  was  its  longitude  and  latitude? 
Ans.    Lon.  254°  9'  14",  Lat.  4°  41'  12"  N. 

Any  nnm-      In  the  following  examples  either  right  ascension  and  decli 
^e  nation  may  be  taken  for  the  data,  and  the  longitude  and  lati 
pies  c&n  be  tude  the  sought   terms,   or   conversely  ;    the  longitude  and 
tound>         latitude  may  be  the  given  data,  and  the  right  ascension  and 


•  As  the  longitude  is  more  than  90°  and  less  than  180°,  the  moon  it 
in  the  second  quadrant  of  right  ascension,  and  77°  53'  in  longitude 
from  the  equator;  and  as  her  latitude  is  south,  it  does  not  correspond 
to  B  S  in  the  figure,  and  we  give  the  example  to  exercise  the  judg- 
ment of  the  learner. 


PRACTICAL   PROBLEMS  54$ 

declination  the  required  terms.     A  Nautical  Almanac  will     £**»•  *• 
furnish  any  number  of  similar  examples. 

R.  A.        Dec.  Lon.         Lat. 

h.  m.  s.     °    '     "  °     '  '    "     °  '     " 

4  15  47  36   15  58  15  south,       238  14  48   4  30  17  north. 

5  6  13  22   18  23   2  north,         93  10  55   5   4  23  south. 

6  112444     145  28  north,        1711240    152  51  south, 

7  20  23  33   14 11    9  south,       304  47  15   5    2  23  north! 

PROBLEM    II. 

Given  the  latitude  of  the  place,  and  the  declination  of  the  sun    Tables  fol 
or  star  ;  to  find  the  semidiurnal  arc,  or  the  time  the  sun  or  star  urnal  aro  an'd 
would  remain  above  the  horizon;  and  to  find  its  amplitude,  or  the  amplitudes 
number  of  degrees  from  the  east  and  west  points  of  the  horizon,  J**^™^1** 
where  it  will  rise  and  set.  lem. 

To  illustrate  this  problem  we  draw  Figure  50.     Let  P  Z  ff,     These  ex. 

&c.,  represent  the  celes- Fig.  50. noTtlke  rt° 

tial     meridian     passing •'^:^•^cVy^^/»^•^E^•v^;>^.:'•;^;;,;4'v^^;B fraction  into 
through  the  place.  Ma 
the  arc  Q  Z  equal  to  the 
latitude,   then    ZP   will] 
equal     the    co-latitude. 
The   line  Hh    is  every- 1 
where  90°  from  Z,  and] 
represents   the    horizon. 
Pp  represents  the  earth'  F 
axis,  and   the  meridian 
90°  distant  from  the  me- 1 
ridian  of  the  place;  Q} 

is  the  equator.  From  the  points  Q  and  q  set  off  d  and  d', 
equal  to  the  declination  (north  or  south,  as  the  case  may  be) 
and  describe  the  small  circle  of  declination,  d  Q  d',  where  this 
circle  crosses  the  circle  of  the  horizon  Hh  is  the  point  where 
the  body  ( sun,  moon,  or  star  )  will  rise  or  set  ( rise  on  one 
side  of  the  meridian  and  set  on  the  other,  both  are  repre- 
sented by  the  same  point  in  the  projection  ).  Through  P  Q 
p  describe  the  meridian  as  in  the  figure,  and  the  right-angled 
spherical  triangle  R  O  0  appears ;  right  angled  at  R. 


246  ASTRONOMY. 

jn  the  triangle  R  Q  C,  there  is  given  the  side  72  Q» 
the  declination,  and  the  angle  opposite  ft  C  Q,  which  is  equal 
to  the  co-latitude.  R  C,  expressed  in  time,  at  the  rate  of  15° 
to  one  hour,  will  be  the  time  before  and  after  6  hours,  from 
the  time  the  body  is  on  the  meridian  to  the  time  it  is  in  the 
horizon ;  and  the  arc  C  Q  is  the  amplitude.  The  triangle  is 
immediately  resolved  by  equations  26  and  27. 

(27.)  Sin.  R  C  =  tan.  declin.  X  tan.  lat. 

sin.  declin. 


(26.,  Sin.  CO 


cos.  lat.    ' 


Observing  that  the  tangent  of  the  latitude  is  the  same  as 
the  cotangent  of  the  angle  R  C  Q,  and  the  cosine  of  the  lati- 
tude is  the  same  as  the  sine  of  R  C  Q,  corresponding  to  a  in 
the  equation. 

EXAMPLE. 

In  the  latitude  of  40°  N.,  when  the  sun's  declination  is  20° 
s  is,  N.,  what  time  before  and  after  six  will  it  rise  and  set,  and  what 
of  course,  ap-  win  be  its  amplitude? 

parent,      be- 
cause   it    re-  (27.)  (26.) 

!o\f '.It           20°       tan'  9-561066  sin-  9.534052 

»nd  not  "*»  40          tan.  9.923813         cos.  9.884254 

8lock-  17°  47'  sin.  9.484879         26°  31'  sin.  9.649798" 

Thus  we  find  that  the  arc  called  the  ascensional  difference^ 
is  17°  47',  or,  in  time,  Ih.  llm.  8s.,  showing  that  the  sun  01 
heavenly  body,  whatever  it  may  be  (when  not  affected  by 
parallax  or  refraction  ),  will  be  found  in  the  horizon  7h.  llm. 
8s.  before  and  after  it  comes  to  the  meridian. 

Its  amplitude  for  that  latitude  and  declination  is  26°  31' 
north  of  east,  or  north  of  west,  and,  if  observed  by  a  compass, 
the  apparent  deviation  would  be  the  variation  of  the  compass. 

2.  At  London,  in  Lot.  51°  32'  N.,  the  sun's  amplitude  wa* 
observed  to  be  39°  48'  toward  the  north ;  what  was  its  declina- 
tion, and  what  was  the  apparent  time  of  its  rising  and  setting  / 
Ans.  Sun's  declination,  23°  27'  59  '  N. 

Sun's  rising,  3h.  47ra.  32s. ;  sun's  setting,  8h.  12m.  28s. 


PRACTICAL  PROBLEMS.  247 

The  amplitude  of  the  sun  is  frequently  observed,  at  sea,  to     CHAP  l 
discover  the  variation  of  the  compass  ;  but,  by  reason  of  re-    r~~ 

J  Refractioi 

fraction,  the  results  are  not  perfectly  accurate.  not  taken  in. 


From  the  right-angled  spherical  triangle  (Fig.  50)  PZQ,  *° 
we  can  Compute  the  time  when  the  sun  is  east  or  west  in  po-  time  that'  the 
sition,  and  the  altitude  it  must  have,  when  in  that  position.  sun  w°uld 
The  angle  Z  is  a  right  angle,  P  Z  is  the  co-latitude,  and  ™JJ*™  the 
is  the  co-declination.  horizon 


Equation  (23)  gives  the  cosine  of  Z  Q,  or  the  sine  of  the  would  **  in' 
altitude  of  the  sun  when  it  is  east  or  west  —  the  latitude  and  while  u  rose 
declination  being  given  —  and  equation  (  24  )  will  give  the  itt  altitude 

i  ,-         f  33'  of  are. 

angle  or  time  trom  noon. 

We  may  also  find  the  altitude  and  azimuth  of  the  sun,  at 
6  o'clock,  by  making  use  of  a  triangle  formed  by  drawing  a 
vertical  through  Z  s  N'.  C  S,  the  given  declination,  will  be  its 
hypothenuse  and  P  Ch,  the  latitude,  will  be  the  arc  of  its 


By  means  of  right-angled  spherical  trigonometry,  as  com- 
prised in  the  equations  from  20  to  33,  we  can  resolve  all  pos- 
sible problems  that  can  occur  in  astronomy,  pertaining  to  the 
sphere;  but,  for  the  sake  of  brevity,  mathematicians,  in  some 
eases,  use  oblique-angled  spherical  trigonometry,  which  is 
nothing  more  than  right-angled  trigonometry  combined  and 
condensed. 

PROBLEM   III. 

Given,  the  latitude  of  t/te  place  of  observation,  the  sun's  de-     The  sun'* 
clination,  and  Us  attitude  above  the  horizon,  to  find  its  meridian  dlstance 

.  -  from  *«  «ne- 

distance,  or  the  time  from  apparent  noon.  ridian,      a. 

There  is  no  problem  more  important  in  astronomy  than  m«a«ured 
that  of  time.     No  astronomer  puts  implicit  faith  in  any  chro-  ar°™         * 


nometer  or  clock,  however  good  and  faithful  it  may  have  and  on  the 
been  ;  and  even  to  suppose  that  a  chronometer  runs  true,  it  e?uat<"  as  a 

circumfer- 

can  only  show  time  corresponding  to  some  particular  me-  ence,  is  the 
ridian;  and  hence,  to  obtain  local  time,  we  must  have  some  measure  of 
method,  directly  or  indirectly,  of  finding  the  sun's  distance  parent  n0on. 
from  the  meridian. 

When  the  center  of  the  sun  is  on  any  meridian,  it  is  than 
tnd  there  apparent  noon  ;  and  the  equation  of  time  will  be  the 


248 


ASTRONOMY 


CHAP.  I. 

Grvat  im- 
portance of 
this  problem. 


Direct  me- 
ridian obser- 
vations not 
generally  ac- 
cnrate. 


Proper  times 
of  observa- 
iica 


Description 
of  the  figure. 


interval  to  or  from  mean  noon;  but  none,  save  an  astronomci 
in  an  observatory,  can  define  the  instant  when  the  sun  is  OB 
the  meridian ;  no  one  else  has  a  meridian  line  sufficiently  defi- 
nite and  accurate,  and  with  him  it  is  the  result  of  great  care, 
combined  with  a  multitude  of  nice  observations. 

To  define  the  time,  then  (when  anything  like  accuracy  ii 
required ),  we  must  resort  to  observations  on  the  sun's  al- 
titude. 

It  is  evident  that  the  altitude  of  the  sun  is  greater  and 
greater  from  sunrise  to  noon,  and  from  noon  to  sunset  the  al- 
titude is  continually  becoming  less.  If  we  could  determine, 
by  observation,  exactly  when  the  sun  had  the  greatest  alti- 
tude, that  moment  would  be  apparent  noon ;  but  there  is  a 
considerable  interval,  some  minutes,  before  and  after  noon,  that 
it  is  difficult  to  determine,  without  the  nicest  observations, 
whether  the  sun  is  rising  or  falling ;  therefore,  meridian  ob- 
servations are  not  the  most  proper  to  determine  the  time. 

From  two  to  four  hours  before  and  after  noon  ( depending 
in  some  respects  on  the  latitude ),  the  sun  rises  and  falls  most 
rapidly ;  and,  of  course,  that  must  be  the  best  time  to  fix 
upon  some  definite  instant ;  for  every  minute  and  second  of 
altitude  has  its  corresponding  time  from  noon ;  and  thus  the 

time  and  altitude  have 
a  scientific  connection, 
which  can  only  be  disen- 
tangled by  spherical  tri- 
gonometry. But  we 
proceed  to  the  problem. 
Draw  a  circle,  P  Z 
Q  //,  &c.,  ( Fig.  51), 
representing  the  meri- 
dian ;  Z  is  the  zenith, 
jand  Z  N  is  the  prime 
vertical ;  Hh  is  the  ho- 
rizon; Z  Q  is  an  arc 
equal  to  the  given  lati- 
tude ;  Q  q  is  the  equa- 
tor, and,  at  right  angles  to  it,  we  have  the  earth's  axis,  P  £ 


Fig.  51. 


PRACTICAL  PROBLEMS.  249 

Take  Ha,  ha,  equal  to  the  observed  altitude  of  the  sun,  CHAP.  i. 
and  draw  the  small  circle,  a  a,  parallel  to  the  horizon,  H h. 
From  the  equator  take  Q  d,  qd,  equal  to  the  declination  of 
the  sun,  and  draw  the  small  circle,  d  d,  parallel  to  Q  q. 
Where  these  two  small  circles,  aa}  dd,  intersect,  is  the  posi- 
tion of  the  sun  at  the  time. 

From  Z  draw  the  vertical,  Z  Q  &,  and  from  P  draw  the 
meridian  through  the  sun,  P  Q  S.  The  triangle  P  Z  Q 
has  all  its  sides  given,  from  which  the  angle  Z  P  Q  can  be 
computed;  which  angle,  changed  into  time  at  the  rate  of  15° 
to  one  hour,  will  give  the  time  from  noon,  when  the  altitude 
was  taken.  If  the  time,  per  watch,  should  agree  with  the 
time  thus  computed,  the  watch  is  right,  and  as  much  as  it 
differs  is  the  error  of  the  watch. 

The  side  Z  Q    is  the  complement  of  the  altitude,  P  O     The  °1>seT 
is  the  complement  of  the  declination,  and  P  Z  is  the  comple-  fines      and 


ment  of  the  latitude,  and  equation  (  35  )  or  ( 36  )  will  solve  points  out  a 
the  problem ;  that  is,  findi  the  angle  P  which  can  be  made  tnang  e* 
to  correspond  to  A,  in  the  equation.     But,  in  place  of  using 
the  complement  of  the  latitude,  we  may  use  the  latitude  it- 
self; and,  in  place  of  using  the  complement  of  the  altitude, 
we  may  use  the  altitude  itself;  provided  we  take  the  cosine, 
when  the  side  of  the  triangle  calls  for  the  sine ;  for  it  would 
be  the  same  thing.     By  thus  taking  advantage  of  every  cir- 
cumstance,   ingenious    mathematicians    have    found    a   less 
troublesome  practical  formula  than  either  (35)  or  (36)  would     Mathema. 
be :  but  we   cannot  stop   to   explain   the   modifications  and  tlc'ans  ma  e 

great      exer 

changes   in  a  work  like  this;  we  contemplate  doing  so  in  tions  to  ab- 
a  work  more  appropriate  to  such  a  purpose :  the  student  must  breviate 
be  content  with  the  following  practical  rule,  to  find  the  time  rations. 
of  day,  from  the  observed  altitude  of  the  sun,  toe/ether  uith  its 
polar  distance,  and  the  latitude  of  the  observer. 

RULE  1. — Add  together  the  altitude,  latitude,  and  polar  dis-       Prectic*. 
tance,  and  divide  the  sum  by  two.     From  this  half  sum  subtract  ™  * 
the  altitude,  reserving  tJie  remainder. 

2. — Take  the  arithmetical  complement  of  the  cosine  of  the  lati- 
tude, the  arithmetical  complement  of  the  sine  of  the  polar  distance, 
the  cosine  of  tfie  half  sum,  and  the  sine  of  the  remainder.  Add 


260  ASTRONOMY. 

CHAP-  '•    these  four  logarithms  together,  and  divide  the  sum  by  two;  the 
result  is  the  logarithmetic  sine  of  half  the  hourly  angle. 

3.— This  angle,  taken  from  the  Tables,  and  converted  into 
time  at  the  rate  of  four  minutes  to  one  degree,  will  be  the 
time  from  apparent  noon ;  the  equation  of  time  applied,  will 
give  the  mean  time  when  the  observation  was  made.* 

*  The  instrument  for  taking  alti- 
tudes at  sea,  or  wherever  the  observer 
may  happen  to  be,  is  a  quadrant  or 
sextant,  according  to  the  number  of 
degrees  of  the  arc.  It  is  made  on  the 
principle  of  reflecting  the  image  of  one 
body  to  another,  by  means  of  a  small 
mirror  revolving  on  a  center  of  motion, 

carrying  an  index  with  it  over  the  arch.  Nearly  opposite 
to  the  index  mirror  is  another  mirror,  one  half  silvered,  the 
other  half  transparent,  called  the  horizon  glass.  Directly  op- 
posite to  the  horizon  glass  is  the  line  of  sight,  in  which  line 
there  is  sometimes  placed  a  small  telescope.  The  line  of 
sight  must  be  parallel  to  the  plane  of  the  instrument.  The 
two  mirrors  must  be  perpendicular  to  the  plane  of  the  instru- 
ment. To  be  in  adjustment,  the  two  mirrors,  namely  the  in- 
dex glass  and  horizon  glass,  must  be  parallel,  when  the  index 
stands  at  0. 

To  examine  whether  a  sextant  is  in  adjustment  or  not, 
proceed  as  follows : 

1.  Is  tJte  index  mirror  perpendicular  to  the  plane  of  the  in* 
strument  ? 

Put  the  index  in  about  the  middle  of  the  arch,  and  look 
into  the  index  mirror,  and  you  will  see  part  of  the  arch  re- 
flected, and  the  same  part  direct;  and  if  the  arch  appears 
perfect,  the  mirror  is  in  adjustment ;  but  if  the  arch  appeari 
broken,  the  mirror  is  not  in  adjustment,  and  must  be  put  BO 
by  a  screw  behind  it,  adapted  to  this  purpose. 

2.  Are  the  mirrors  parallel  when  the  index  is  at  0  ? 

Place  the  index  at  0,  and  clamp  it  fast;  then  look  at  some 
well-defined,  distant  object,  like  an  even  portion  of  the  dis- 


PRACTICAL  PROBLEMS 


251 


EXAMPLE. 

In  latitude  39°  46'  north,  when  the  sun's  declination  was 
3°  27'  north,  the  altitude  of  the  sun's  center,  corrected  for 
refraction,  index  error,  &c.,  was  32°  20',  nsing ;  what  was 
the  apparent  time? 

20 

-  cos.  comple.      -      .114268 

-  sine  comple.      -      .000788 

10         -        9  .267652 


CHAP.  I 


Altitude, 
Latitude, 
Polar  dis., 

32 
39 

86 
2)158" 

20 
46 
33 
~39 

30 

79 
32 

19 
20 

46 

59 

30 

Z  P         24    50 


30  sine 

2 


9  .864090 


.246798 


9  .623399 


The  hourly  angle  is  49  41  0,  which,  converted  into  time, 
gives  3h.  18m.  44s.,  the  time  from  apparent  noon,  and,  as 

tant  horizon,  and  see  part  of  it  in  the  mirror  of  the  horizon 
glass,  and  the  other  part  through  the  transparent  part  of  the 
glass ;  and,  if  the  whole  has  a  natural  appearance,  the  same 
as  without  the  instrument,  the  mirrors  are  parallel;  but,  if 
the  object  appears  broken  and  distorted,  the  mirrors  are  not 
parallel,  and  must  be  made  so,  by  means  of  the  lever  and 
screws  attached  to  the  horizon  glass. 

3.  Is  the  Jiorizon  glass  perpendicular  to  the  plane  of  the  in- 
strument ? 

The  former  adjustments  being  made,  place  the  index  at  0, 
and  clamp  it ;  look  at  some  smooth  line  of  the  distant  horizon, 
while  holding  the  instrument  perpendicular ;  a  continued,  un- 
broken line  will  be  seen  in  both  parts  of  the  horizon  glass ; 
and  if,  on  turning  the  instrument  from  the  perpendicular,  the 
horizontal  line  continues  unbroken,  the  horizon  glass  is  in  full 
adjustment;  but,  if  a  break  in  the  line  is  observed,  the  glass 
is  not  perpendicular  to  the  plane  of  the  instrument,  and  must 
be  made  so,  by  the  screw  adapted  to  that  purpose 

After  an  instrument  has  been  examined  according  to  these 


£52  ASTRONOMY. 

CHAP.  i.    the  sun  was  rising,  it  was  before  noon,  and  the  apparent  time 

was  8  h.  41  m.  16  s. 
An  are  may      ^  good  observer,  with  a  eood  instrument,  in  favorable  cir- 

tx»  measured 

by  the  quad-  cumstances,  can  define  the  time,  from  the  sun  s  altitude,  to 

rant    within  within  three  or  four  seconds. 

°An "artificial  ^i  sea>  *^e  observer  brings  the  reflected  image  of  the  sun 
to  the  horizon,  and  allows  for  the  dip  ( Tables  p.25).  On  shore, 
where  no  natural  horizon  can  be  depended  upon,  resort  is  had 
to  an  artificial  horizon,  which  is  commonly  a  little  mercury 
turned  out  into  a  shallow  vessel,  and  protected  from  the  wind 
by  a  glass  roof.  The  sun,  or  any  other  object,  may  be  seen 
reflected  from  the  surface  of  the  mercury  (  which,  of  course, 
is  horizontal ),,  and  the  image,  thus  reflected,  appears  as  much 
below  the  natural  horizon  as  the  real  object  is  above  the  hori- 
zon; and,  therefore,  if  we  measure,  by  the  instrument,  the 
angle  between  the  object  and  its  image  in  the  artificial  hori- 
zon, that  angle  will  be  double  the  altitude. 

When  mercury  is  not  at  hand,  a  plate  of  molasses  will  do 
very  well;  and  in  still,  calm  weather,  any  little  standing  pool 
of  water  may  be  used  for  an  artificial  horizon. 

Observations  taken  in  an  artificial  horizon  are  not  affected 
by  dip,  but  they  must  be  corrected  for  refraction  and  index 
error,  and,  if  the  two  limbs  of  the  sun  are  brought  together, 
its  semidiameter  must  be  added  after  dividing  by  two. 

A  practical      The  following  example  is  from  a  sailor's  note  book  : 

«On  the  lgth  of  May>   1848>  at  gea>  in  latitude  36o  21 

north,  longitude  54°  10'  west,  by  account,  at  7  h.  43  m.  pei 
watch ;  the  altitude  of  the  sun's  lower  limb  was  32°  51'  ris- 
ing; the  hight  of  the  eye  was  eighteen  feet,  and  the  index 

directions,  it  may  be  considered  as  in  an  approximate  adjust- 
ment— a  re-examination  will  render  it  more  perfect — and, 
finally,  we  may  find  its  index  error  as  follows : — measure  the 
sun's  diameter  both  on  and  off  the  arch — that  is,  both  ways 
from  0,  and  if  it  measures  the  same,  there  is  no  index  error ; 
but  if  there  is  a  difference,  half  that  difference  will  be  the  in- 
dex error,  additive,  if  the  greatest  measure  is  off  the  ardi, 
sub  tractive,  if  on  the  arch. 


PRACTICAL  PROBLEMS. 


error  of  the  sextant  was  2'  12" 
ror  of  the  watch?" 


additive.     What  was  the  er- 


253 
.L 


7  h.  43  m.,  morning. 
3     38 


PREPARATION. 

Time,  per  watch, 

Longitude,  54°  10',  in  time, 

Estimated  mean  time  at  Greenwich,  11  h.  21  m. 

The  declination  of  the  sun  at  mean  noon,  Greenwich  time, 
was  19°  38'  29"  increasing,  the  daily  variation  being  13' ; 
the  variation,  therefore,  for  39',  the  time  before  noon,  was 
21"  subtractive.  Hence,  the  declination  of  the  sun,  at  the 
time  of  observation,  was  19°  38'  8"  north,  and  the  polar  dis- 
tance 70°  21'  52". 

Observed  altitude, 

Index  error,  ... 

Semidiameter,        ... 

Refraction,     -       ... 

Dip  of  the  horizon, 

True  altitude  of  sun's  center, 

Altitude,      33°    3' 20" 


Preparation* 
to  be    mad* 
according  to 
circum- 
stance*. 


32°  51'  00" 
+      2  12 
+    15  49 

—  1  28 

—  4  13 

33°    3' 20" 


Latitude, 
Polar  dis., 


36 
70 


21 
21 


52 


2)139    46  12 


cos.  complement, 
sin.  complement, 


69 
33 


53 
3 


6 

20 


36    49  46 


cosine,     - 


sine, 


.093982 
.026013 

•      9.536470 

9.777770 
2)19.434235 
9.717117 


£  hourly  angle,  31    25  30         sine, 
This  angle  corresponds  to  4h.  llm.  24s.,  or,  in  reference 

to  the  forenoon,  7  h.  48  m.  36  s.  apparent  time. 

On  the  18th  of  May,  noon,  Greenwich  time,  the  equation 

of  time  was  3  m.  54  s.  subtractive ;  therefore,  the  true  mean 

time,  at  ship,  was        -  -         7  h.  44  m.  42  s. 

Time,  per  watch,     -         -         -         7      43 
Watch  slow,  -        -  1     42 

A  short  time  before  this  observation  was  taken,  the  watch 


Bjr  obser- 
rations  thni 
taken  at  dif- 
ferent time* 
at  the  same 
place,  the 
rate  of  the 
watch  can  be 
determined. 


254  ASTRONOMY. 

CHAP.I.  was  compared  with  a  chronometer  in  the  cabin,  which  was 
too  fast  for  mean  Greenwich  time,  19  m.  12.5  s.,  according  to 
estimation  from  its  rate  of  motion.  The  chronometer  was 
fast  of  watch  by  3  h.  56  m.  39  s.  What  was  the  longitude  of 
the  ship? 

h.    m.    •. 

Time  of  observation,  per  watch,  7  43  00 
Diff.  between  watch  and  chron.,  3  56  39 
Time,  per  ch.,  at  observation,  11  39  39 
Chron.  fast  of  Greenwich  time.  19  12 

Greenwich  mean  time,  -  11  20  27 
Mean  time  at  ship,  -  -  7  44  42 

Longitude  in  time,      -  3  35  45=53°  56'  west. 

Howtode.  The  longitude  is  west,  because  it  is  later  in  the  day,  at 
Greenwich,  tnan  at  tne  ship-  This  example  explains  all  the 


whether  the  details  of  finding  the  longitude  by  a  chronometer. 

lonjitade  is      -g    taking  advantage  of  the  observations  for  time  on  shore, 

east  or  west.  e 

Howtode-  we  may  draw  a  meridian  line  with  considerable  exactness; 
termine  and  for  instance,  in  the  last  observation  (  if  it  had  been  on  land  ), 
-  24  s.  after  the  observation  was  taken,  the  sun 


line.  would  be  exactly  on  the  meridian  ;  and  if  the  watch  could  be 

depended  upon  to  measure  that  interval  with  tolerable  accu- 
racy, the  direction  from  any  point  toward  the  sun's  center, 
at  the  end  of  that  interval,  would  be  a  meridian  line.  Sev- 
eral such  meridians,  drawn  from  the  same  point,  from  time  to 
time,  and  the  mean  of  them  taken,  will  give  as  true  a  me- 
ridian as  it  is  practical  to  find  ;  although,  for  such  a  purpose, 
a  prominent  fixed  star  would  be  better  than  the  sun. 
Absolute  The  problem  of  time  includes  that  of  longitude,  and  find- 
ing the  difference  of  longitude  between  two  places  always  re- 
solves itself  into  the  comparison  of  the  local  times,  at  the  same 
instant  of  absolute  time.  When  any  definite  thing  occurs, 
wherever  it  may  be,  that  is  absolute  time.  For  instance, 
the  explosion  of  a  cannon  is  at  a  certain  instant  of  absolute 
time,  wherever  the  cannon  may  be,  or  whoever  may  note  the 
event  ;  but  if  the  instant  of  its  occurrence  could  be  known 
at  far  distant  places,  the  local  clocks  would  mark  very  diffe- 


PRACTICAL    PROBLEMS.  265 

rent  hours  and  minutes  of  time ;  but  such  difference  would  be     cmr  I 
occasioned  entirely  by  difference  of  longitude :  the  event  is 
the  same  for  all  places  —  it  is  &  point  in  absolute  time. 

Thus  any  single  event  marks  a  point  in  absolute  time.  If  Absolute 
the  same  event  is  observed  from  different  localities,  the  diffe-  by  meang  Of 
rence  in  local  time  will  give  the  difference  in  longitude.  But  events, 
a  perfect  clock  is  a  noter  of  events,  it  marks  the  event  &  ^Octe°rc  0" 
of  noon,  the  event  of  sunrise,  the  event  of  one  hour  after  events,  when 
noon,  &c. ;  and  if  we  could  have  perfect  confidence  in  this  u  rons  trae| 

butnototner 

marker  of  events,  nothing  more  would  be  necessary  to  give  us  wjge. 
the  local  time  at  a  distant  place.  The  time,  at  the  place 
where  we  are,  can  be  determined  by  the  altitude  of  the  sun, 
or  a  star,  as  we  have  just  seen.  But,  unfortunately,  we  can- 
not have  perfect  confidence  in  any  chronometer  or  clock ;  and 
therefore  we  must  look  for  some  event  that  distant  observers 
can  see  at  the  same  time. 

The  passage  of  the  moon  into  the  earth's  shadow  is  such   Eclipses  an 
an  event,  but  it  occurs  so  seldom  as  to  amount  to  no  practical  J^*^*'  maik 
value.     The  eclipses  of  Jupiter's  satellites  are  such  events,  absolute 
but  they  cannot  be  observed  without  a  telescope  of  consider-  time' but  fo1 

»  L  common  pur- 

able  power,  and  a  large  telescope  cannot  be  used  at  sea.  poses     they 
Hence  these  events  are  serviceable  to  the  local  astronomer  are  of  little 
only ;  the  sailor  and  the  practical  traveler  can  be  little  bene- 
fited by  them.     The  moon  has  comparatively  a  rapid  motion 
among  the  stars  (  about  13°  in  a  day  ),  and  when  it  comes  to 
any  definite  distance  to  or  from  any  particular  star,  that  cir- 
cumstance may  be  called  an  event,  and  it  is  an  event  that  can 
be  observed  from  half  the  globe  at  once. 

Thus,  if  we  observe  that  the  moon  is  30°  from  a  particular  The  motion 
star,  that  event  must  correspond  to  some  instant  of  ^absolute  among  the 
time ;  and  if  we  are  sufficiently  acquainted  with  the  moon,  stars,  may  be 
and  its  motion,  so  as  to  know  exactly  how  far  it  will  be  from  ^"a^dex 
certain  definite  points  (  stars )  at  the  times,  when  it  is  noon,  moving 
3,  6,  9,  &c.,  hours  at  Greenwich,  then,  if  we  observe  these 
events  from  any  other  meridian,  we  thereby  know  the  Green-  soiate  tim. 
wich  time,  and,  of  course,  our  longitude. 

Finding  the  Greenwich  time  by  means  of  the  moon's  angu- 
lar distance  from  the  sun  or  stars,  is  called  taking  a,  lunar; 


256  ASTRONOMY. 

CHAF.  i.    and  it  is  probably  the  only  reliable  method  for  long  voyages 
at  sea. 

If  the  motion  of  our  moon  had  been  very  slow,  or  if  the 
earth  had  not  been  blessed  with  a  moon,  then  the  only 
methods,  for  sea  purposes,  would  have  been  chronometers  and 
dead  reckoning.  For  a  practical  illustration  of  the  theory  of 
lunars,  we  mention  the  following  facts. 

•e^atioUn".      In  &  86a  Journal  of  1823>  'li  is  stated  that  the  distance  of 

Uutrated  by  the  moon  from  the  star  Antares  was  found  to  be  66°  37'  8", 

an  exampi*.  V)}ien  f/te  observation  was  properly  reduced  to  the  center  of  tJie 

earth,   and  the  time  of  observation  at  ship  was  September 

16th,  at  7h.  24m.  44s.  p.  M.  apparent  time. 

By  comparing  this  with  the  Nautical  Almanac,  it  was 
found  that  at  9  P.  M.,  apparent  time  at  Greenwich,  the  dis- 
tance between  the  moon  and  Antares  was  66°  5'  2",  and  at 
midnight  it  was  67°  35'  31";  but  the  observed  distance  was 
between  these  two  distances,  therefore  the  Greenwich  time 
was  between  9  and  12  p.  M.,  and  the  time  must  fall  between 
9  and  12  hours  in  the  same  proportion  as  66°  37'  8"  falls 
between  the  distances  in  the  Nautical  Almanac;  and  thus  an 
observer,  with  a  good  instrument,  can  at  any  moment  deter- 
mine the  Greenwich  time,  whenever  the  moon  and  stars  are 
in  full  view  before  him. 

The  moon,  in  connection  with  the  stars  in  the  heavens, 
may  be  considered  a  public  clock  (  quite  an  enlargement  of 
the  town-clock  ),  by  which  certain  persons,  who  understand 
the  dial  plate  and  the  motion  of  the  index,  and  who  have  the 
necessary  instrument,  can  read  the  Greenwich  time,  or  the 
time  corresponding  to  any  other  meridian  to  which  the  com- 
putations may  be  adapted. 

observed      The  angular  distances  from  the  moon  to  the    sun,  stars, 
irtances^  and  planets,  as  put  down  in  the  Nautical  Almanac,  corre- 
tancei      ai  spending  to  every  third  hour,  are  distances  as  seen  from  the 
soen     from  cen^er  Of  ^ne  earth,  and  when  observations  are  taken  on  th 
Die  earth,      surface  the  distance  is  a  little  different ;  the  position  of  flu 
moon  is  affected  by  parallax  and  refraction,  the  sun  or  stai 
^J  refraction  alone ;  and  therefore  a  reduction  is  necessary, 
which  is  called  clearing  the  distance.     This  is  done  by  spheri- 


PROPORTIONAL    LOGARITHMS.  257 

cal  trigonometry.  The  distance  between  the  moon  and  star  CBAP.  I. 
is  observed,  the  altitudes  of  the  two  bodies  are  also  observed. 
The  co-altitudes  come  to  the  zenith,  and  the  co-altitudes, 
with  the  distance,  form  three  sides  of  a  spherical  triangle, 
from  which  the  angle  at  the  zenith  can  be  computed.  Then 
correct  the  altitude  of  the  moon  for  parallax  and  refraction, 
and  the  star  for  refraction,  and  find  the  true  altitudes  and  co- 
altitudes,  and  the  true  co-altitudes  and  angle  at  the  zenith 
give  two  sides  and  the  included  angle  of  a  spherical  triangle, 
and  the  third  side,  computed,  is  the  true  distance. 

An  immense  amount  of  labor  has  been  expended  by  mathe- 
maticians, to  bring  in  artifices  to  abbreviate  the  computation 
of  clearing  lunar  distances ;  and  they  have  been  in  a  measure 
successful,  and  many  special  rules  have  been  given,  but  they 
would  be  out  of  place  in  a  work  of  this  kind. 

PROPORTIONAL     LOGARITHMS. 

In  every  part  of  practical  astronomy  there  are  many  pro- 
portional  problems  to   be  resolved,  and  as   the   terms  are  'og 
mostly  incommensurable,  it  would  be  very  tedious,  in  most  tjon  of  y,. 
cases,  to  proceed  arithmetically,  we  therefore  resort  to  loea-  construction 

•*1  J  1          t   1  -^  •      Of  *•     **bl« 

nthms,  and  to  a  prepared  scale  of  logarithms,  very  appropri-  giren> 
atcly  called  proportional  logarithms. 

The  tables  of  proportional  logarithms  commonly  correspond 
to  one  hour  of  time,  or  60'  of  arc,  or  to  three  hours  of  time. 
The  table  in  this  book  corresponds  to  one  hour  of  time,  or 
3600  seconds  of  either  time  or  arc.  To  explain  the  construc- 
tion and  use  of  a  table  of  proportional  logarithms,  we  propose 
the  following  problem  : 

At  a  certain  time,  the  moon's  hourly  motion  in  longitude  was 
33'  17" ;  how  much  would  it  change  its  longitude  in  13m.  23s.  ? 

Put  x  to  represent  the  required  result,  then  we  have  the 
following  proportion : 

m.      m.    s.  '     " 

60  :  13  23  :  :  33  17  :  x\ 
Or  3600  :  13  23  :  :  33  17  :  *. 

Divide  the  first  and  second  terras  of  this  proportion  by  the 
17 


258 


ASTRONOMY. 


L  second,  and  the  third  and  fourth  by  the  third,  then  we  have 
3600  x 

13.23   :  :   33.17 


Divide  the  third  and  fourth  terms  by  x,  and  multiply  the 
same  terms  by  3600,  and  the  proportion  becomes 

3600  a       3600    (    3600 

13.23   :  x       !   33.17' 


Multiplying  extremes  and  means,  using  logarithms,  and  re 
membering  that  the  addition  of  logarithms  performs  multipli- 
cation, 

3600       .       /3600\   ,  .       /3600\ 
Then  we  have  log.  —  —  =  log.  (-^  +log.  (^--). 


By  the  construction  of  the  table,  the  proportional  logarithm 


of  1"  is  the  common  logarithm  of 


S600 


;  of  2"  is  the  com- 


mon  logarithm  of 


3600  .    3600 

—  ;  of  3     is  — 


3600 
,  &c.,  to  ; 


1  hence  the  proportional  logarithm  of  3600  is  0. 

,1          11 
We  now  work  the  problem  : 


13  23 
33  17 


-  -    -    P.  L. 

-  -    -    P.  L. 


6516 
2559 


Result,   -  -25i  -    -    -    P.  L.    9075 

Examples  EXAMPLES    FOR    PRACTICE. 

lustrate  the      1.  When  the  sun's  hourly  motion  in  longitude  is  2'  29", 
fa  change  Of  longitude  in  37  m.  12  s.? 

- 

AnS.  1    62    .5. 


practical  nti- 

lity  of  proper. 
tional  logar- 


2.  When  the  moon's  decimation  changes  57".2  in  one  hour, 
what  will  it  change  in  17  m.  31  s.  ?  Ans.  16".8. 

3.  When  the  moon  changes  longitude  27'  31"  in  an  hour, 
how  much  will  it  change  in  7  m.  19  s.  ?  Ans.  3'  21". 

4.  When  the  moon  changes  her  right  ascension  1  m.  58  s. 
in  one  hour,  how  much  will  it  change  in  13  m.  7  s.  ? 


• 


Ans.  25".8. 


PROPORTIONAL    LOGARITHMS.  259 

N.  B.  This  table  of  proportional  logarithms  will  solve  any     CHAP.  i. 
proportion,  provided  the  first  term  is  60,  or  3600 ;  therefore, 
when  the  first  term  is  not  60,  reduce  it  to  60,  and  then  use 
the  table. 

EXAMPLES. 

1.  If  the  sun's  declination  changes  16'  83"  in  twenty-four     Exampiet 
hours,  what  will  be  the  change  in  14  h.  18m.?  fiven  to  "' 

Instrate     the 

Statement,     24    :     14.18     ::     16'  33"  practical  a*. 

hty  of  propor- 
Or,  12      :         7.09  tional   Iog9r 

Or,  60    :    35.45    :  :    16'  33"  ithn"- 

16' 33"    P.  L.    5594 
35'  45"    P.  L.    2249 


Ans.    9'  51".5  P.  L.     7843 

2.  If  the  moon  changes  her  declination  1°  31'  in  twelve 
hours,  what  will  be  the  change  in  7  h.  42  m.  ?        Ans.  58'. 

Conceive  degrees  and  minutes  to  be  minutes  and  seconds, 
and  hours  and  minutes  to  be  minutes  and  seconds. 

When  60  minutes  or  3600  seconds  are  not  the  first  term  of 
a  proportion,  the  result  is  found  by  taking  the  difference  of 
the  proportional  logarithms  of  the  other  term  for  the  P.  L. 
of  the  sought  term,  as  in  the  following  example : 

The  moon's  hourly  motion  from  the  sun  is  26'  30",  what 
time  will  it  require  to  gain  30"  ? 

Statement,  26'  30"  :  60m.  :  30"  :     x  other 

30"        P.  L.        2.0792 
60  m.      P.  L.        0.0000 

Product  of  extremes,  2.0792 

26'  30"  P.  L.  sub.    3549 

Besult,          1m.  7  s.     P.  L.     1.7243 

3.  The  equation  of  time  for  noon,  Greenwich,  on  a  certain 
day,  was  6  m.  21  s. ;  the  next  day,  at  noon,  it  was  6  m.  43  s. : 
what  was  it  corresponding  to  3  h.  42  m.  p.  M.,  in  longitude 
74°  west,  on  the  same  day  ?  Ang.  6  m.  29  B. 


?60  ASTRONOMY. 


CHAPTER   II. 

GENERAL   PROBLEM. 

CHAP.  ii.  Given,  the  motions  of  sun  and  moon,  to  determine  their  appa- 
A  general  reni  positions  at  any  given  time  ;  from  which  results  their  appa- 
problem  pre-  rent  relative  situations,  and  the  eclipses  of  the  sun  and  moon. 
thecmnpnta^  ^^s  problem  covers  two- thirds  of  the  science  of  astronomy, 
tioE«ofeciip.  and  includes  many  minor  problems ;  therefore  a  brief  or  hasty 
"•*•  solution  must  not  be  expected. 

From  the  foregoing  portions  of  this  work,  the  reader  is 
supposed  to  have  acquired  a  good  general  knowledge  of  the 
solar  and  lunar  motions,  and  the  tables  give  all  the  particu- 
lars of  such  motions;  and  all  the  artifices  and  ingenuity  that 
astronomers  could  devise,  have  been  employed  in  forming  and 
arranging  these  tables,  for  the  double  purpose  of  facilitating 
the  computations  and  giving  accuracy  to  the  results. 

The  tables,  in  general,  must  be  left  to  explain  themselves, 
and  the  mere  heading,  combined  with  the  good  judgment  of 
the  reader,  will  furnish  sufficient  explanation,  in  most  in- 
stances ;  but  some  of  them  require  special  mention.  All  (he 
tables  are  adapted  to  mean  time  at  Greenwich. 

EXPLANATION    OF    TABLES. 

,  A  very  ge-      Table  IV  contains    the  sun's  mean  longitude,  the  longi- 
tude of  its  perigee  (each  diminished  by  2°),  and  the  Argu- 
ana.  ments  *  for  some  of  the  small  inequalities  of  the  sun's  appa- 
rent motion. 


tion    of    the 
tables. 


Explanation  *  The  term,  ARGUMENT,  in  astronomy,  means  nothing  more  than  a 
•f  the  term  correspondence  in  quantities.  Thus,  each  and  every  degree  of  the 
gun's  longitude  corresponds  with  a  particular  amount  of  declination  { 
and  therefore,  a  table  could  be  formed  for  the  declination,  and  the  ar- 
gument would  be  the  sun's  longitude. 

The  moon's  horizontal  parallax  and  semidiameter  vary  together, 
and  each  minute  of  parallax  corresponds  to  a  particular  amount  of  se- 
midiameter; hence,  a  table  can  be  made  for  finding  the  semidiameter, 
and  the  arguments  would  be  the  horizontal  parallax.  But  the  hori- 


EXPLANATION  OF    TABLES.  261 

Argument  I,  corresponds  to  the  action  of  the  moon;  Ar-  CHAP.  li. 
gument  II,  to  the  action  of  Jupiter;  Argument  III,  to  Ve- 
nus ;  and  Argument  N,  is  for  the  equation  of  the  equinoxes, 
and  corresponds  with  the  position  of  the  moon's  node ;  and, 
by  inspecting  the  column  in  the  table,  it  will  be  perceived 
that  the  argument  runs  round  the  circle  in  a  little  more  than 
eighteen  years,  as  it  should;  and  thus,  by  inspection,  we  can 
obtain  an  insight  as  to  the  period  of  any  argument  in  the 
eolar  or  lunar  tables. 

The  object  of  diminishing  the  mean  longitude  and  perigee  Explanation 
of  the  sun  by  2°,  is  to  render  the  equation  of  the  center  al-  of  the  sola* 
ways  additive ;  for  if  2°  are  taken  from  the  longitude,  and  2° 
added  to  the  equation  of  the  center,  the  combination  of  the 
two  quantities  will  be  the  same  as  before ;  and,  as  the  equa- 
tion of  the  center  is  always  less  than  2°,  therefore,  2°  added 
to  its  greatest  minus  value,  will  give  a  positive  result.  By 
the  same  artifice  all  equations  may  be  rendered  always  posi- 
tive. The  2°,  taken  from  the  mean  longitude,  are  restored  by 
adding  1°  59'  30"  to  the  equation  of  the  center,  and  10"  to 
each  of  the  other  equations ;  hence,  to  find  the  real  equation 
of  the  center  corresponding  to  any  degree  of  the  anomaly, 
subtract  1°  59'  3"  from  the  quantity  found  in  the  table. 

Table  XI,  shows  the  time  of  the  mean  new  moon,  &c., 
in  January,  diminished  by  fifteen  hours,  to  render  the  correc- 
tions always  additive.  The  fifteen  hours  are  restored  by  add- 
ing 4h.  20m.  to  the  first  equation,  10  h.  10m.  to  the  second, 
10  m.  to  the  third,  and  20  m.  to  the  fourth. 

Argument  I,  corrects  for  the  action  of  the  sun  on  the  lunar 

zontal  parallax  and  semidiameter  of  the  moon  depend  (not  solely)  on  the 
moon's  distance  from  its  perigee;  hence,  a  table  can  be  formed  giving 
both  horizontal  parallax  and  semidiameter;  which  ARGUMENTS  are  the 
anomaly.  In  other  words,  an  argument  may  be  called  an  INDEX,  and 
when  the  arguments  correspond  to  points  in  a  circle,  or  to  the  differ- 
ence of  points  in  a  circle,  the  circle  may  be  considered  as  divided  into 
1000  or  100  parts,  then  500,  or  50,  as  the  case  may  be,  would  corre- 
spond to  half  a  circle,  and  so  on  in  proportion.  This  mode  of  dividing 
the  circle  had  been  adopted,  with  certain  limitations,  to  avoid  the 
greater  labor  of  computing  by  denominate  numbers. 


ASTRONOMY. 

CHAI-.  ii.    orbit ;  Argument  II,  corrects  for  the  mean  eccentricity  of  the 
lunar  orbit ;  Argument  III,  corrects  for  the  different  combina- 
tions of  the  solar  and  lunar  perigee ;  and  Argument  IV,  cor- 
1        rects  for  the  variation  occasioned  by  the  inclination  of  the 
lunar  orbit  to  the  ecliptic ;  N.  shows  the  distance  from  or  to 
the  nodes. 
Tables  ad-      New  and  full  moons,  calculated  by  these  tables,  can  be  de- 

•tno'dicV  *  Pende(l  upon  within  four  minutes,  and  commonly  much  nearer; 

motion  of  the  but  when  great  accuracy  is  required,  the  more  circuitous  and 

moon,      by  eiaij0rate  method  of  computing  the  longitudes  of  both  sun 

which      new 

and        full  and  moon  must  be  employed. 

moons  can  be      Tables  XIII,  XIV,  and  XV,  are  used  in  connection  with 
•°mpnteA     Table  XL 

Explanation      Table  XVI,  shows  the  reduction  of  the  latitude,  and  also  of 
table  *ne  moon's  horizontal  parallax,  corresponding  to  the  latitude, 

occasioned  by  the  peculiar  shape  of  the  earth,  and  the  dimi- 
nution of  its  diameter  as  we  approach  the  poles.  The  table 
is  put  in  this  place  because  of  the  convenient  space  in  the  page. 

Table  XVII,  and  the  following  tables  to  No.  XXX,  contain 
the  arguments  and  epochs  of  the  moon's  mean  longitude,  erec- 
tion, &c.,  necessary  in  computing  the  moon's  true  place  in 
the  heavens. 

rhe  method      The  argument  for  evection  is  diminished  by  29';  the  ano- 
*  malJ  bJ  IQ  59'»  the  variation  by  8°  59',  and  the  longitude 
e  by  9°  44',  and  the  balances  are  restored  by  adding  the  same 
amounts  to  the  various  equations,  which,  at  the  same  time, 
renders  the  equation  affirmative,  as  explained  in  the  solar 
tables. 

The  arguments  in  Table  xxxn, are  also  arguments  for  polar 
distance,  or  latitude,  in  Table  XXTIII.  Anything  like  a  minute 
explanation  of  these  tables  would  lead  us  too  far,  and  not 
comport  with  the  design  of  this  work.  The  use  of  the  tables 
will  be  shown  by  the  examples. 

We  have  carried  the  mean  motions  of  the  sun  and  moon 
only  to  five  minutes  of  time  —  and  this  is  sufficient  for  all 
practical  purposes  —  for  we  can  proportion  to  any  interne 
diate  minute  or  second,  by  means  of  the  hourly  motions. 


PRACTICAL  PROBLEMS.  263 


PROBLEM    I. 

From  the  solar  tables  find  the  sun's  longitude,  hourly  motion 
in  longitude,  declination,  semidiameter  and  equation  of  time; 
and  for  a  specific  example,  find  these  elements  corresponding  to 
mean  time,  at  Greenwich,  1854,  May  26  d..  8  h.  40  m. 

To  find  the  sun's  declination,  spherical  trigonometry  gives 
us  the  following  proportion  :  (Eq.  20,  page  231.) 

As  radius  -        -          10.000000 

Is  to  sin.  of  Q's  Ion.  (65°  12'  15")  -  -  9.957994 
So  is  sin.  of  obliq.  of  the  eclip.  (  23°  27'  32")  9.599900 
To  sin.  declination  N.,  21°  10'  54"  -  -  9^557894 

In  nearly  all  astronomical  problems,  time  is  reckoned  from 
noon  to  noon  —  from  0  hour  to  24  hours. 

When  the  given  time  is  apparent,  reduce  it  to  mean  time, 
and  when  not  at  Greenwich,  reduce  it  to  Greenwich  time,  by 
applying  the  longitude  in  time.  —  (  This  is  necessary  because 
the  tables  are  adapted  to  Greenwich  mean  time.  ^ 

From  Table  IV,  and  opposite  the  given  year,  take  out  the 
whole  horizontal  line  of  numbers  (  headed  as  in  the  table  ) 
and  from  Tables  V,  VII,  VIII,  take  out  the  numbers  corre- 
sponding to  the  month  —  day  of  the  month  —  hour  and 
minute  of  the  day,  as  in  the  following  example. 

Add  up  the  perpendicular  columns,  as  in  compound  num-     The  snn's 
bers,  rejecting  entire  circles  in  every  column,  and  the  sums  or  duta^ces 
surplus,  as  the  case  may  be,  will  give  the  mean  values  of  all  gee  point  u 
the  quantities  for  the  given  instant.  called  itf 

Subtract  the  longitude  of  the  perigee  from  the  mean  Ion-  m9^t 
gitude,  and  the  remainder  will  be  the  mean  anomaly  ;  which  is 
the  argument  for  the  equation  of  the  center. 

With  the  respective  arguments  take  out  the  corresponding 
equations,  all  of  which  add  to  the  mean  longitude,  and  the 
true  longitude  of  the  sun  from  the  mean  equinox  will  be  found. 

With  the  argument  N*  take  out  the  equation  of  the  equi- 

•  The  reason  why  N  is  not  applied  with  the  other  equation*  te  be- 
cause it  ia  sometimes  negative. 


264 
<?IU 


ASTRONOMY. 


noxes  from  Table  X,  and  apply  it  according  to  its  sign,  and 
the  result  will  be  the  true  longitude  from  the  true  equinox. 


1854 
Eq.  < 

May 

26  d 
8h 
40m 

rf  cent 
I 
I] 
I 

M.  Lon. 

Lon.  Perig. 

I. 

II. 

III. 

N. 

8.     o    '      a 

9    84848 
3  28  16  40 
24  38  28 
1943 
-  139 

S.     0     '      a 

9    82529 
20 
4 

073 
59 
844 
11 

998 
301 
63 
0 

902 
206 
43 
0 

809 
18 
4 
0 

987    362    151    831 
9    82553 
2    2    518 

4  23  39  25  =  Mean  anomaly. 

Sun's  hourly  motion  in  Ion.  2'  24" 
"       semidiameter,            15'  49' 

2    2  5  18 
er    3  6  42 
10 
[              13 
[I             8 

2   51231 
Eq.  of  the  equinox  —  16 

True  Ion.        2    5  12  15 

Th»»»    rin- 
ciples     were 
explain**/  on 
7 ages  94 
ud  05. 


To  find  the  Agnation  of  time  to  great  accuracy. 

By  equation  21,  page  231,  we  find 

the  sun's  R.  A., 

Subtract  this  from  the  sun's  Ion.,     - 
Equatorial  point  is  west  of  mean  east- 

ward motion  by 

From  the  equation  of  the  center,  as 

just  found, 

Subtract  the  constant  of  the  table, 
The  sun  east  of  its  mean  place,         - 
Subtract  (  b  )  from  (  a  )  because  one 
is  east,  the  other  west,  and  we 
have  the  arc  48'  53" 

This  arc,  converted  into  time,  gives  3m.  15.5s.  for  the 
equation  of  time  at  this  instant,  and  the  sun  will  not  come  to 
the  meridian  at  mean  noon,  but  3m.  15£s.  afterward, 
Hence,  to  convert  mean  into  apparent  time,  in  the  month  of 
May,  add  the  equation  of  time. 


-  63  16  10 

-  65  12  15 

-  1°  56'  5"  (a> 

-  3  6  42 

-  1  59  30 
1 


7  12 


PRACTICAL    PROBLEMS.  265 

Thus,  in  general,  we  can  determine  the  exact  amount  of  CHAF.  IL 
the  equation  of  time,  by  means  of  the  two  arcs  (  a  )  and  (  b  ) 
( which  are  roughly  tabulated  on  page  95 ),  and,  without 
strictly  scrutinizing  each  particular  case,  we  can  determine 
whether  we  are  to  take  the  sum  or  difference  of  the  arcs  by 
inspecting  the  table  on  page  95,  or  by  referring  our  results  to 
some  respectable  calendar. 

EXAM  PLE. 

2.  What  will  be  the  sun's  longitude,  declination,  right  as- 
cension, hourly  motion  in  longitude,  semidiameter  of  the  su: , 
and  equation  of  time  corresponding  to  20  minutes  past  9, 
mean  time  at  Albany,  N.  Y.,  on  the  17th  of  July,  1860  V 

N.  B.  At  this  time  the  sun  will  be  eclipsed. 

Ans.  Lon.  114°  38'  21    ;  Dec.  21°  12'  48". 

R.  A.,  in  time,  7h.  46m.  15s. ;  Eq.  of  time  to  add  to  apparent 
time,  5m.  46.2s.;  hourly  motion  in  Ion.,  2'  23";  S.  D.,  15'  45.6' . 

PROBLEM    II. 

From  Tables  XI,  XII,  and  XIII,  to  find  the  approximate  time 
of  new  and  full  moons. 

Take  the  time  of  new  moon,  and  its  arguments,  from  Table 
XI,  corresponding  to  January  of  the  given  year,  and  take 
as  ma  iy  lunations,  from  the  following  table,  as  correspond  to 
the  number  of  the  months  -after  January,  for  which  the  new 
moon  is  required;  add  the  sums,  rejecting  the  sums  corre- 
sponding to  whole  circles,  in  the  arguments,  and  in  the  column 
of  days,  rejecting  the  number  corresponding  to  the  expired 
months,  as  indicated  by  Table  XIII;  the  sums  will  be  the 
mean  new  moon  and  arguments  for  the  required  month. 

When  a  full  moon  is  required,  add  or  subtract  half  a  luna-  Add  th« 
tion.  Sometimes  one  more  lunation  than  the  number  of  the  numberofln- 
month  after  January,  will  be  required  to  bring  the  time  to  cessary  t« 
the  required  month,  as  it  occasionally  happens  that  two  luna-  brins  the  re- 
tions  occur  in  the  same  month.  quired  ^im*. 

Apply  the  equations  corresponding  to  the  different  argu-  of  year, 
ments  taken  from  Table  XIV,  and  their  sum,  added  to  the 
mean  time  of  new  or  full  moon,  will  give  the  true  mean  time 
of  new  or  full  moon  for  the  meridian  of  Greenwich,  within 
four  minutes,  and  generally  within  two  minutes. 


266 


ASTRONOMY. 


CHAP.  IL        For  the  time  at  any  other  meridian  apply  the  time  corre- 
sponding to  the  longitude. 

EXAMPLES. 

1.  Required  the  approximate  time  of  new  moon,  in  May. 
1854,  corresponding  to  the  day  of  the  month,  and  the  time  of 
the  day,  at  Greenwich,  England,  Boston,  Mass.,  and  Cincin- 
nati, Ohio. 


January. 

Mean   N.  Moon. 

I. 

II.    I 

III. 

IV.)    N. 

1854, 
Four  Luna. 

Table  XIII. 

27d.  18h.  14m. 
118      2    56 

0761 
3234 

1168 
2869 

19 

61 

04  1  668 
96  ]  341 

145    21     10 
120 

3995 

4037 

80 

00  |  009 

N  shows  an  eclipse  of  the 
sun  —  visible  in  the  United 
States. 

May, 
J.. 
II. 
III. 
IV. 

25     21     10 
6    46 
4    14 
17 

-    20 

May, 

26      8    47 

-      4 


New  C>  mean  time  at  Greenwich,      -      8  h.  47  m.,  p.  M. 

Boston,  Lorigitude,          -  4     44 

New  f)  Boston  time, 

Cincinnati,  Longitude  from  Boston, 

New  €)  Cincinnati  time, 

2.  Required  the  approximate  time  of  full  moon,  in  Jufy, 
1852,  for  the  meridian  of  Greenwich,  and  for  Albany  time, 
New  York. 


January. 

Mean  N.  Moon. 

I. 

II. 

III. 

IV. 

N. 

1852, 
Five  Luna. 
Half  Luna. 

20d.  llh.  53m. 
147     15    40 
14    18     22 

0549 
4042 
404 

3239 
3586 
5359 

38 
76 

58 

27 
95 
50 

538 
426 
43 

Tab.  13.  Bis. 

182    21     55 

182 

4995 

2184 

72 

72 

007 

The  column  N  shows  that 
the  moon   is  very  near  her 
node.    There  will  be  a  total 
eclipse  of  the  moon  —  invisi- 
ble in  the  United  States. 

Mean  time  at  Greenwich. 

July, 

IL 
III. 
IV. 

0    21     55 
4    21 
42 
17 
10 

1      8    25 

July, 

ECLIPSES.  267 

Full  0  Greenwich  time,        -        -        3  h.  25  m.  p.  M.        CHA».  a 

Albany,  Longitude,       -  4 55 

Full  m  Albany  time,     -  -       10      30       A.  M. 

Thus  we  can  compute  tbe  time  of  new  or  full  moon  for  any 
month  in  any  year ;  but,  as  the  numbers  for  the  arguments 
correspond  to  mean  or  average  motions,  and  cannot,  without 
immense  care  and  labor,  be  corrected  for  the  true,  variable 
motions,  the  results  are  but  approximate,  as  before  observed. 

ECLIPSES. 

Eclipses  take  place  at  new  and  full  moons ;  an  eclipse  of  When  «cHp- 
the  sun  at  new  moon,  and  an  eclipse  of  the  moon  at  full  ***c<> 
moon;  but  eclipses  do  not  happen  at  every  new  and  full 
moon;  and  the  reason  of  this  must  be  most  clearly  compre- 
hended by  the  student  before  it  will  be  of  any  avail  for  him  to 
prosecute  the  further  investigation  of  eclipses. 

If  the  moon's  orbit  coincided  with  the  ecliptic,  that  is,  if     wh-T  "^ 

sea    do    not 

the  moon's  motion  was  along  the  ecliptic,  there  would  be  an  take     p]ace 
eclipse  of  the  sun  at  every  new  moon,  and  an  eclipse  of  the  every  month 
moon  at  every  full  moon ;  but  the  moon's  path  along  the  ce- 
lestial   arch    does    not    coincide   with   the   sun's    path,    the 
ecliptic  ;  but  is  inclined  to  it  by  an  angle  whose  average  value 
is  5°  8',  crossing  the  ecliptic  at  two  opposite  points  on  the 
apparent  celestial  sphere,  which  are  called  the  moon's  nodes. 

If  the  moon's  path  were  less  inclined  to  the  ecliptic,  there  What  would 
would  be  more  eclipses  in  any  given  number  of  years  than  j£  **""^*[ 
now  take  place.  If  the  moon's  path  were  more  inclined  to  whatforfew. 
the  ecliptic  than  it  now  is,  there  would  be  fewer  eclipses.  er  ecliP9e> 

The  time  of  the  year  in  which  eclipses  happen,  depends  on 
the  position  of  the  moon's  nodes  on  the  ecliptic;  and  if  that 
position  were  always  the  same,  the  eclipses  would  always 
happen  in  the  same  months  of  the  year.  For  instance,  if  the 
longitude  of  one  node  was  30°,  the  other  would  be  in  longi-  why  a. 
tude  30+180,  or  210°;  and,  as  the  sun  is  at  the  first  of  eclips* 

should    tak« 

these  points  about  the  20th  of  April,  and  at  the  second  about  p]ac*  in  any 
the  20th  of  October,  the  moon  could  not  pass  the  sun  in  P»rtlcnl« 

month. 

these  months  without  coming  very  nearly  in  range  with  it,  of 
oouro^,  producing  eclipses  in  April  and  October 


ASTRONOMY. 

Fig.  52.  For  a  clearer  illustration,  we 

present  Fig.  52:  the  right  line 
through  the  center  of  the  figure, 
represents  the  equator,the  curved 
line  qposrcz,  crossing  the  equa- 
tor at  two  opposite  points,  re- 
presents the  ecliptic;  and  the 
curved  line  Q  O  Q  represents 
the  path  of  the  moon  crossing 
the  ecliptic  at  the  points  Q  and 
Q;  the  first  of  these  points  is 
the  descending,  the  other,  the  as- 
cending node. 

As  here  represented,  the  as- 
cending node  is  in  longitude 
about  210°,  and  the  descending 
node  in  about  30°;  which  was 
about  the  situation  of  the  nodes 
in  the  year  1846,  and,  of  course, 
the  eclipses  of  that  year  must 
have  been,  and  really  were,  in 
April  and  October. 

The  sun  and  moon  at  con- 
junction  are  represented  in  the 


nas  passed  the  northern  tropic, 

whjch  mugt  be  ab(mt  the  firgt  of 
»  .     .. 

August;  and  it  is  perfectly  evi- 
dent that  no  eclipse  can  then 
take  place,  the  moon  running 
past  the  sun,  at  a  distance  of 
about  Jive  degrees  south  ;  and  at 
the  opposite  longitude,  the  moon 
must  pass  about  Jive  degrees 
north. 

The  moon's  nodes  move  back- 
ward at  the  mean  rate  of  19° 
19'  per  year;  but  the  sun  moves 


ECLIPSES. 

over  19°  in  about  twenty  days ;  therefore,  the   eclipses,   on    CHA*«  n. 
an  average,  must  take  place  about  twenty  days  earlier  each 
year,  or  at  intervals  of  about  346  days. 

In  May,  1846,  the  moon's  ascending  node  was  in  longi- 
tude 216° ;  in  eight  years,  at  the  rate  of  19°  19'  per  year, 
it  would  bring  the  same  node  to  longitude  61°  28'.  The  sun 
attains  this  longitude  each  year  on  the  23d  of  May;  there- 
fore, the  eclipses  for  1854  must  happen  in  May,  and  in  the 
opposite  month,  November. 

In  computing  the  time  of  new  and  full  moons,  as  illustrated    The  mean' 
by  the  preceding  examples,  the  columns  marked  N,  not  hith-  Ium°s  N.Tn 
erto  used,  indicate  the  distance  of  the  sun  and  moon  from  the  table» 
the  moon's  node   at  the  time  of  conjunction  or  opposition.     • 

The  circle  is  conceived  to  be  divided  into  1000  parts,  com-    Eclipses  are 
mencing  at  the  ascending  node ;  the  other  node  then  must  limited  to  a 
be  at  500 ;  and  when  the  moon  changes  within  37  of  0,  or  along      th« 
500,  that  is,  37  of  either  node,  there  must  be  an  eclipse  of  ecliPtir 
the  sun,  seen  from  some  portion  of  the  earth.     When  the 
distance  to  the  node  is  greater  than  37,  and  less  than  53, 
there  may  be  an  eclipse,  but  it  is  doubtful :  we  shall  explain 
how  to  remove  the  doubt  in  the  next  chapter. 

When  the  moon  fulls  within  25  divisions  of  either 
node,  there  must  be  an  eclipse  of  the  moon :  when  the  dis- 
tance is  greater  than  25,  and  less  than  35,  the  case  is 
doubtful ;  but,  like  the  limits  to  the  new  moon,  the 

Trr  .     .  .      Comparative 

doubts   are    easily  removed.      We  repeat,   the   ecliptic  limits  number     of 
for  eclipses  of  the  sun  are  53  and  37  ;  for  eclipses  of  the  moon,  «eiip«et    »f 
the  limits  are  35  and  25.     Hence,  in  any  long  period  of  time,  moon> 
the  number  of  eclipses  of  the  sun  is,  to  the  number  of  eclipses 
of  the  moon,  as  53  to  35. 

In  the  same  period  of  time,  say  in  one  hundred  years,  there 
will  be  more  visible  eclipses  of  the  moon  than  of  the  sun ;  for 
every  eclipse  of  the  moon  is  visible  over  half  the  world  at 
once,  while  an  eclipse  of  the  sun  is  visible  only  over  a  very 
small  portion  of  the  earth ;  therefore,  as  seen  from  any  one 
place,  there  are  more  eclipses  of  the  moon  than  of  the  sun. 
,  In  the  preceding  examples  the  columns  N  are  far  within 
the  limits,  and,  of  course,  there  must  be  an  eclipse  of  the 


270  ASTRONOMY. 

CHAP,  ii.    gun  on  the  26th  of  May,  1854,  and  an  eclipse  of  the  moon  in 

July,  1852. 

HOW  we  As  N  is  in  value  9,  at  the  time  of  new  moon,  in  May,  1854, 
eclipse  of  the  ^  shows  that  the  moon  will  then  have  passed  the  ascending 
«nn  will  hap.  node,  and  be  north  of  the  ecliptic,  and  the  eclipse  must  be 
aeth  o^Maij,  v^ble  on  the  northern  portions  of  the  earth,  and  not  on  the 

1854,        and  southern. 

^™mgt^*  When  the  moon  changes  in  south  latitude,  which  will  be 
we  learn  that  shown  by  N  being  a  little  more  than  500,  or  a  little  less  than 
HiTse1*  to  1^0,  ^e  corresponding  eclipse,  if  of  the  sun,  will  be  visible 
some  north.  on  some  southern  portion  of  the  earth,  and  not  visible  in  the 
em  portion  of  northern  portion;  and  if  of  the  moon,  the  moon  will  run 
through  the  southern  portion  of  the  earth's  shadow. 

Table B,p.31, shows  the  moon's  latitude,  approximately  cor- 
What  indi-  responding  to  the  column  N ;  or  N  is  the  argument  for  the 
catet  that  a  latitude,    and   the   heading  of   the   argument    columns   will 
will  *>•  vh».  show  whether  the  moon  is  ascending  to  the  northward,  or  de- 
We  *»a  t«me  wending  to  the  southward. 

le  The  tables  from  XVI  to  XVIII,  together  with  the  solar 
tables,  will  give  approximate  values  of  the  elements  necessary 
for  the  calculation  of  eclipses  ;  and  if  accurate  results  are  not 
expected,  these  tables  are  sufficient  to  present  general  princi- 
ples, and  give  primary  exercises  to  the  student  in  calculating 
eclipses ;  but  he  who  aspires  to  be  an  astronomer,  must  con- 
tinue the  subject,  and  compute  from  the  lunar  tables,  far- 
ther on. 

The  times,  and  the  intervals  of  time,  between  eclipses,  de- 
pend on  the  relative  motion  of  the  sun  and  moon,  and  the 
motion  of  the  moon's  nodes.  The  relative  motion  of  the  sun 
and  moon  is  such  as  to  bring  the  two  bodies  in  conjunction, 
or  in  opposition,  at  the  average  interval  of  29  d.  12  h.  44  m. 
3  s.,  and  the  retrograde  motion  of  the  node  is  such  as  to  bring 
the  sun  to  the  same  node  at  intervals  of  346  d.  14  h.  52  m. 
16  e.  Neglecting  the  seconds,  and  conceiving  the  sun,  moon, 
and  node  to  be  together  at  any  point  of  time,  and  after  an  un- 
known interval  of  time,  which  we  represent  by  P,  sup- 

p 
pose  them  together  again.     Then  ^     ^  represents  the 


ECLIPSES.  2~j 

number  of  returns  of  the  lunation  to  the  node  m  the  time    CHAP.  11. 
P,  and  the  expression          14.   RO    rePresen*s  ^ne  number  of  of  the   snn 

and  moon  in 

returns  of  the  sun  to  the  node  in  the  same  time.  Eacn  re-  relation  to 
turn  of  either  body  to  the  node  is  unity  ;  therefore,  these  ex-  mooi»'»  >»®d» 
pressions  are  to  each  other  as  two  whole  numbers  ;  say  as  m  m 

to  *;  that  is,  —  ^-g  :         r  :  :  m  :  »; 


Or, 


(29  12  44)     (346  14  52)' 
Or,       -       (346  14  52)rc=(29  12  44)m    -     -    -     (a) 

n  __  29  12  44 
'  f»~346  14  52* 

Reducing  to  minutes,  and  dividing  numerator  and  denomi- 

n      10631 

nator  by  4,  we  have  — =  .     As  this  last  fraction  is  ir- 

reducible, and  as  m  and  n  must  be  whole  numbers  to  answer 
the  assumed  condition,  therefore,  the  smallest  whole  number 
for  m  is  124783,  and  for  n  is  10631;  that  is,  as  we  see  by 
equation  (  a  ),  the  sun,  moon,  and  node  will  not  be  exactly  to- 
gether a  second  time,  until  a  lapse  of  124783  lunations,  or 
10631  returns  of  the  sun  to  the  same  node ;  which  require  a 
period  of  no  less  than  10088  years  and  about  197  days.  We 
say  about,  because  we  neglected  seconds  in  the  computation, 
and  because  the  mean  motions  will  change,  in  some  slight  de- 
gree, through  a  period  of  so  long  a  duration. 

This  period,  however,  contemplates  an  exact  return  to  the    THU  period 
same  positions  of  the  sun,  moon,  and  earth,  so  that  a  line  p^S'Tm! 
drawn  from  the  center  of  the  sun  to  the  center  of  the  moon  possibilities, 
would  strike  the  earth's  axis  in  exactly  the  same  point ;  but 
to  produce  an  eclipse,  it  is  not  necessary  that  an  exact  return  Exactcoin. 
to   former    position    should  be   attained;  a  greater  or  less  cide nee. ne». 
approximation  to  former  circumstances  will  produce  a  greater  er  haPP«n- 
or  less  approximation  to  a  former  eclipse :  but  exact  coinci- 
dences, in  all  particulars,  can  never  take  place,  however  long 
the  period. 

To  determine  the  time  when  a  return  of  eclipses  may  hap- 


*72  ASTRONOMY. 

CHAP'  fl-   pen  (  particularly  if  we  reckon  from  the  most  favorable  posi- 

HOW     to  tions  —  that  is,  commence  with  the  supposition  that  the  sun, 

io4  the  $uc-  moon   an(}  node  are  together  ),  it  is  sufficient  to  find  the  first 

eessive      ie- 

lorn     of  ]  0631 

approximate  values  of  the  fraction  VoTob*      ^  we  nn(^  tne 


successive   approximate  fractions,  by  the   rule  of  continued 
fractions,*  we  shall  have  the  successive  periods  of  eclipses, 
which  happen  about  the  same  node  of  the  moon 
The  approximating  fractions  are 

_1         J.          _3          4_  19_          JL56 

Tl        12         36        47          223"          1831* 

These  fractions  show  that  11  lunations  from  the  time  an 
•bowing  the  ec^Pse  occurs,  we  may  look  for  another;  but  if  not  at  11,  it 
period*  at  will  be  at  12,  and  it  may  be  at  both  11  and  12  lunations; 
**lKh  and  at  five  or  six  lunations,  we  shall  find  eclipses  at  the  other 

eclipse*     oc- 

cur. node,  and  the  same  succession  of  periods  occurs  at  both 

nodes.  ^ 

To  be  more  certain  of  the  time  when  an  eclipse  will  occur, 
we  must  take  35  lunations  from  a  preceding  eclipse,  which 
period  is  1033  days  13  h.  40  m.,  and  the  sun  at  that  time  is 
about  6°  40'  farther  from,  or  nearer  to,  the  node,  than  before 
—  and,  if  the  count  is  from  the  ascending  node,  the  moon's 
latitude  is  about  38'  farther  south  than  before;  and  if  from 
the  descending  node,  the  moon  is  about  the  same  distance 
farther  north. 

The  double  of  11,  12,  and  35  lunations,  from  any  eclipse, 
may  also  bring  an  eclipse. 

If  an  eclipse  occurs  within  10°  of  either  node,  it  is  certain 

that  eclipses  will  again  happen  after  the  lapse  of  47  lunations. 

A  brief  ex-      The  period  of  47  lunations  includes  1387  d.  22  h.  31m., 

and  4  rcvolutions  of   the  sun  to  the  node   include  1386  d. 
of  H  h.  29m.;    the  difference  is  1  day  11  h.  29m.;    but  in  this 
eoiip*e«.       time  the  sun  will  move,  in  respect  to  the  node,  1°  32   and 
some  seconds  ;  therefore,  if  the  first  eclipse  were  exactly  at  the 
node,  the  one  which  follows  at  the  expiration  of  47  lunations, 

•See  Robinson's  Arithmetic. 


ECLIPSES.  273 

or  3  years  and  nearly  11  months  afterward,  would  take  place    CH*».  *i. 
1°  32'  short  of  the  same  node ;   and  if  it  were  the  ascending 
node,  the  moon's  latitude  would  be  about  8'  40"  south,  and 
if  the  descending  node,  about  8'  40"  more  to  the  north. 

The  period,  however,  which  is  most  known,  and  the  most 
remarkable,  appears  in  the  next  term  of  the  series,  which 
shows  that  223  lunations  have  a  very  close  approximate  value 
to  19  revolutions  of  the  sun  to  the  node. 

The  period  of  223  lunations  includes  6585.32  days,  and  19 
returns  of  the  sun  to  the  same  node  require  6585.78  days, 
showing  a  difference  of  only  a  fraction  of  a  day ;  and  if  the  dis™e  °^l~ 
sun  and  moon  were  at  the  node,  in  the  first  place,  they  would  omen  called 
be  only  about  20'  from  the  node,  at  the  expiration  of  this  ^  j>eriod 
period,  and  the  difference  in  the  moon's  latitude  would  be 
less  than  2',  and  therefore  the  eclipse,  at  the  close  of  this 
period,  must  be  nearly  the  same  in  magnitude  as  the  eclipse 
at  the  beginning;  and  hence  the  expression  "a  return  of  the 
eclipse"  as  though  the  same  eclipse  could  occur  twice. 

This  period  was  discovered  by  the  Chaldaean  astronomers,     By  this  pe. 
and  enabled  them  to  give  general  and  indefinite  predictions  ^akeTtnm" 
of  the  eclipses  that  were  to  happen ;  and  by  it  any  learner,  mary  predic- 
however  crude  his  mathematical  knowledge,  can  designate  the  tic 
day  on  which  an  eclipse  will  occur  from  simply  knowing  the 
date  of  some  former  eclipse.     The  period  of  6585  days  is  18 
years,  including  4  leap  years,  and  11   days  over;  therefore 
from  any  eclipse,  if  we  add  18  years  and  11  days,  we  shall 
come  within  one  day  of  the  time  of  an  eclipse,  and  it  will  be 
an  eclipse  of  about  the  same  magnitude  as  the  one  we  reckon 
from. 

For  the  purpose  of  illustrating  the  method  of  computing    Asnmmarj 
lunar  eclipses,  we  wish  to  find  the  time  when  some  future  ™°ine  °  c°t™J 
eclipse  of  the  moon  will  take  place ;  and  from  the  American  time     when 
Almanac  of  1833,  we  find  that  an  eclipse  of  the  moon  took  *"njt  JjjJjJ"" 
place  on  the  1st  day  of  July  of  that  year,  therefore  "a  re- 
turn of  this  eclipse"  must  take  place  on  the  12th  of  July 
1851. 

By  a  simple  glance  into  the  American  Almanac  for  the 
year  1834,  we  find  a  total  eclipse  of  the  moon  on  the  21st  of 
18 


574  ASTRONOMY. 

CHIP  ii.  June  —  therefore,  on  the  first  of  July  1852,  or  at  the  time 
that  the  moon  fulls  on  or  about  the  first  of  July,  there  must 
be  a  large  eclipse  of  the  moon,  visible  to  all  places  from  where 
the  moon  will  then  be  above  the  horizon;  and  furthermore,  18 
years  and  11  days  after  this,  that  is,  in  the  year  1870,  on  the 
12th  day  of  July,  the  moon  will  be  again  eclipsed;  and,  in 
this  way,  we  might  go  on  for  several  hundred  years,  but  in  time 
the  small  variations,  which  occur  at  each  period,  will  gradu- 
ally wear  the  eclipse  away,  and  another  eclipse  will  as  gradu- 
ally come  on  and  take  its  place. 

In  the  same  manner  we  may  look  at  the  calendar  for  any 
year,  take  any  eclipse,  that  is  anywhere  near  either  node,  and 
run  it  on,  forward  or  backward. 

Let  us  now  return  to  the  eclipse  of  July  12th,  1851. 
Elements      To  decide  all  the  particulars  concerning  a  lunar  eclipse  we 
co™"  must  have  the  following  data,  commonly  called  elements  of 
lunar  the  eclipse : 

ecHpses.  j    ^he  time  of  full  moon. 

2.  The  semidiameter  of  the  earth's  shadow. 

3.  The  angle  of  the  moon's  visible  path  with  the  ecliptic. 

4.  Moon's  latitude. 

5.  Moon's  hourly  motion. 

6.  Moon's  semidiameter. 

7.  The  semidiameter  of  the  moon  and  earth's  shadow 
General  di-      To  find  these  elements,  the  approximate  time  of  full  noon 

^taiTtheeT- is  found  from  Tal)le  XI'  and  tbe  tables  immediately  con- 
ement.8  of  nected.  For  the  time  thus  found,  compute  the  longitude  of 
t^e  gun  £rom  rj^bie  jy?  anj  the  tables  immediately  con- 
nected, as  illustrated  by  examples  on  page  254. 

Compute,  also,  the  latitude,  longitude,  horizontal  parallax 
semidiameter,  and  hourly  motion  in  latitude  and  longitude, 
from  the  lunar  tables,  commencing  with  Table  XVI,  and  fol- 
lowing out  the  computation  by  a  strict  inspection  of  the  ex- 
amples we  have  given  (  rules,  aside  from  the  examples,  would 
be  of  no  avail ) ;  and,  if  the  longitude  of  the  moon  is  exactly 
180°  in  advance  of  the  sun,  it  is  then  just  the  time  of  full 
moon;  if  not  180°,  it  is  not  full  moon;  if  more  than  180°,  it 
is  past  full  moon. 


ECLIPSES.  275 

It  will  rarely,  if  ever,  happen  that  the  longitude  of  the    CHAP.II. 
moon  will  be  exactly  180°  in  advance  of  the  longitude  of  the 
sun ;  but  the  difference  will  always  be  very  small,  and,  by 
means  of  the  hourly  motions  of  the  sun  and  moon,  the  time 
of  full  moon  can  be  determined  by  the  problem  of  the  couriers* 

The  moon's  latitude  must  be  corrected  for  its  variation, 
corresponding  to  the  variation  in  time  between  the  approxi- 
mate and  true  time  of  full  moon. 

To  find  the  semidiameter  of  the  earth's  shadow,  where  the   Rule  to  f  a<l 

the   semidia- 

moon  runs  through  it,  we  have  the  following  rule :  meter  of  the 

To  the  moon's  horizontal  parallax,  add  the  surfs,  and,  from  earth's    sha> 

ike  #tim,  subtract  the  sun's  semidiameter. 

This  rule  requires  demonstration.     Let  S  (Fig.  53)  be 

Fig.  53. 


the  center  of  the  sun,  h  the  center  of  the  earth,  and  Pm  a 
small  portion  of  the  moon's  orbit.  Draw  p  P,  a  tangent  to 
both  the  earth  and  sun  ;  from  p  and  P,  draw  P  E  and  p  E, 
forming  the  triande  p  £  P. 

By   inspecting   the   figure,    we    perceive    that    the    three   Demonstra, 
angles:  '7  of  tbe 


Also,  the  three  angles  of  the  triangle,  P  Ep,  are,  together, 
equal  to  180°; 

Therefore,     SEp+p  E  P+m  EP=P+p+p  EP  ; 

Drop  the  angle,  p  E  P,  from  both  members  of  the  equation, 
and  transpose  the  angle   SEp9  we  then  have 


•  RoU  MOB'S  Algebra  —  problem  of  the  couriers. 


276  ASTRONOMY. 

CHAP-  n-  But  the  angle,  m£P,ia  the  semidiameter  of  the  earth's 
shadow  at  the  distance  of  the  moon;  S  Ep  is  the  semidiame- 
ter of  the  sun ;  P,  that  is,  the  angle  £Pp,  is  the  moon's 
horizontal  parallax ;  and  p  is  the  horizontal  parallax  of  the 
sun ;  therefore,  the  equation  is  the  rule  just  given.* 

The  angle  of  the  moon's  visible  path  with  the  ecliptic  is  al- 
angie  of  the  ways  greater  than  its  real  path  with  the  ecliptic,  and  depends, 
moon's  visi-  jn  8ome  measure,  on  the  relative  motions  of  the  sun  and 

ble  path  with 

&«  ecliptic.     m°0n- 

To  explain  why  the  real  and  visible  paths  of  the  moon  are 
different,  let  A  B  (  Fig.  54 )  be  a  portion  of  the  ecliptic,  and 
A  m  a  portion  of  the  moon's  orbit;  then  the  angle, 

Fig.  54. 

**"   "" 


is  the  angle  of  the  moon's  real  path  with  the  ecliptic.  Con 
ceive  the  sun  and  moon  to  depart  from  the  node,  A,  at  the 
same  time,  the  moon  to  move  from  A  to  m  in  one  hour,  and 
the  sun  to  move  from  A  to  I  in  the  same  time ;  join  b  and  m, 
and  the  angle  mbB  is  the  angle  of  the  moon's  visible  path 
with  the  ecliptic,  which  is  greater  than  the  angle  mAB', 
which  is  the  angle  of  the  moon's  real  path  with  the  ecliptic. 
On  this  principle  we  determine  the  angle  in  question. 

All  the  other  elements  are  given  directly  from  the  tables. 

*  Some  writers  have  directed  us  to  increase  this  value  of  the  shadow 
by  its  cne-sixtlelli  part,  but  we  emphatically  deny  the  propriety  of  the 
direction. 


ECLIPSES. 


277 


CHAPTER    III. 


PREPABATION   FOE   THE  COMPUTATION   OP   ECLIPSES. 

WE   shall  now  go  through  the  computation  in  full,  that  it  CHAP,  m. 
may  serve  for  an  example  to  guide  the  student  in  computing 
other  eclipses. 


Mean  N.  Moon. 

I. 

II. 

111. 

IV. 

N. 

1851, 
Six  Luna. 
Half  Luna. 

Id.  14h.  21m. 
177       4     24 

14    18    22 

0038 
4851 
404 

3916 
4303 
5359 

40 
92 

58 

39 
95 
50 

431 
511 
43 

193    13      7 
181 

5293 

3578 

90 

84 

985 

As  N  is  within  25  of  1000, 
or  0,  there  must  be  an  eclipse. 
The  sun  is  15  short  of  the  as- 
cending node,  and  the  moon  at 
full,  being  opposite,  must  be  15 
short  of  the   descending    node, 
and  therefore,  in  north  latitude, 
descending. 

July, 

II. 
III. 
IV. 

12     13      7 
3     35 
2      9 
14 
11 

Full  9 

12     19     16 

tion  of  a  lu- 
nar eclipse. 

The  approx- 
imate time  of 
fall  moon 
computed 


We  now  compute  the  sun's  longitude,  hourly  motion,  and    Sun»«  ion 
eemidiameter  for  1851,  July  12,  19  h.  15m.  mean  Greenwich  gitud°  cora 

puted,  corre- 

time,  as  follows:  spending  to 

the  approxi- 
mate time  of 
ful  noon. 


1851 

Eq.  ol 

July 
12  d 
19  h 
15m 

f  cente 
I. 
II 
II 

O  M.  Lon. 

Loh.  Peri. 

I. 

II. 

III. 

N. 

8      °     '       " 

9    83239 
52824    8 
10  50  32 
4649 
037 
3183445 
r     1  39  38 
10 
18 
L            20 

S.     O     '      " 

9    8  22  24 
31 

2 

9   82257 
3  18  34  45 

958 
129 
371 
27 

250 
454 

28 
0 

025 
310 
19 
0 

648 
27 
2 
0 

677" 

485 

732 

151 

6  10  11  48  =  Mean  anomaly. 

O  's  hourly  motion,             2'  23" 
O's  semidiameter,             15'  46" 

3201511 
Eq.  of  equinox      —  16 

O  Ion.           3  20  14  55 

278  ASTRONOMY. 

CHAP.  in.       We  now  compute  the  moon's  longitude,  latitude,  semidi- 
Direction  anieter,  horizontal  parallax,  and  hourly  motions  for  the  same 
for   comput-  mean  Greenwich  time,  as  follows : 

ing  the 

moon's    true 
longitude.  FOR     THE     LONGITUDE. 

1.  Write  out  the  arguments  for  the  first  twenty  equations, 
and  find  their  separate  sums.     With  these  arguments  enter 
the  proper  tables  (  as  shown  by  the  numbers  ),  and  take  out 
the  corresponding  equations,  and  find  their  sum. 

2.  Write  out  the  evection,  anomaly,  variation,  longitude, 
supplement  to  node,  and  the  several  arguments  for  latitude, 
in  separate  columns,  corresponding  to  the  given  time,  and 
write  the  sum  of  the  twenty  preceding  equations  in  the  column  of 
evection. 

3.  Add  up  the  column  of  evection  first ;  its  sum  will  be 
the  corrected  argument  of  evection,  with  which,  take  out  the 
equation  of  evection  (  Table  XXIV  ),  and  write  it  under  the 
sum  of  the  first  twenty  equations ;  their  sum  will  be  the  cor- 
rection to  put  in  the  column  of  anomaly. 

4.  Add    up    the  column  of  anomaly,   and    the    sum   will 
be  the  moon's  corrected  anomaly,  which  is  the  argument  for 
the  equation  of  the  center.     With  this  argument  take  out  the 
equation  of  the  center  from  Table  XXV,  and  write  it  under 
the  sum  of  the  preceding  equations,  and  find  the  sum  of  all, 
thus  far.     Write  this  last  sum  in  the  column  of  variation, 
and  then  add  up  the  Column  of  variation ;  which  sum  is  the 
correct  argument  of  variation,  and  with  it  take  out  the  equa- 
tion for  variation  from  Table  XXVII. 

5.  Add  the  equation  for  variation  to  the  sum  of  all  the 
preceding  equations,  and  the  sum  will  be  the  correction  for 
longitude,  which,  put  in  the  column  of  longitude,   and  the 
whole  added  up,  will  give  the  moon's  longitude  in  lier  orbit, 
reckoned  from  the  mean  equinox. 

Equation      6.  Add  the  orbit  longitude  to  the  supplement  of  the  node, 
noils'  so^e"  an<^  tne  suin  *s  *^e  argument  of  reduction  to  the  ecliptic ;  it 
times  called  is  also  the  first  argument  for  polar  distance, 
notation    in      ^j^  tlie  a™^^  Of  reduction  take  out  the  reduction 

'on^itude 

from  Table  XXXII,   and  add  it  to  the  longitude. 


ECLIPSES.  279 

With  argument  19,  which  is  the  same  as  N  in  the  solar  to-  CHAP  ra. 
lies,  take  out  the  equation  of  the  equinox,  and  apply  it  ac- 
cording to  its  sign ;  the  result  will  be  the  moon's  true  longi- 
tude reckoned  on  the  ecliptic  from  the  true  equinox. 

FOR    THE    LATITUDE. 

Add  the  same  correction  ( to  its  nearest  minute )  to  column    General  <h- 
II,  as  was  added  to  the  column  of  longitude,  and  add  its  »<*»<»»•    foi 
value,  expressed  in  the  1000th  part  of  a  circle,  to  all  the  fol-  moon>s  iati. 
lowing  columns,  except  column  X.     Add  up  these  columns,  t«d« 
rejecting  thousands  ( or  full  circles ),  and  the  sums  will  be 
the  5th,  6th,  7th,  8th,  9th,  and  10th  arguments  of  latitude. 

The  sum  of  the  moon's  orbit  longitude,  and  supplement  to 
node,  is  the  first  argument  of  latitude.  The  sum  of  column 
II  is  the  second  argument  of  latitude ;  the  moon's  true  longi- 
tude is  the  third  argument,  and  the  twentieth  of  longitude  is 
the  fourth  argument.  Then  follow  5,  6,  &c.,  up  to  10. 
With  these  arguments  enter  the  proper  Tables,  and  take  out 
the  corresponding  equations,  and  their  sum  will  be  the  moon's 
true  distance  from  the  north  pole  of  the  ecliptic,  and,  of  course, 
will  be  in  north  latitude  if  the  sum  is  less  than  90°,  otherwise 
in  south  latitude. 

N.  B.  When  the  first  argument  of  latitude  is  nearer  6  signs 
than  12  signs,  the  moon  is  tending  south;  when  nearer  12  signs, 
or  0  sign,  than  6  signs,  it  is  tending  north. 

For  tlie  equatorial  horizontal  parallax. —  The  arguments  for  Equator... 
Eveetion,  Anomaly,  and  Variation  are  also  arguments  for  Parallax  and 
horizontal  parallax,  and  with  these  arguments  take  out  the  ter  depend 
corresponding  equations  from  the  tables  adapted  to  this  °P°n  eacb 

other. 

purpose. 

For  the  semidiameter. —  The  equatorial  parallax  is  the  ar- 
gument for  semidiameter,  Table  XXXIV. 

For  the  hourly  motion  in  longitude. —  Arguments  2,  3,  4,  and  General  di- 
5  of  longitude  sensibly  affect  the  moon's  motion ;  they  are,  Jj^j™8  ^ 
therefore,  arguments  for  hourly  motion,  Table  36  ( the  units  hourly  mo- 
and  tens  in  the  arguments  are  rejected ).  Take  out  these  tion  of  lh* 
equations  from  table,  also  take  out  the  equation  correspond- 
ing to  the  argument  of  evection,  Table  XXXVII  With  the 


t30  ASTRONOMY. 


CHAF.  ill.  sum  of  the  preceding  equations,  at  the  top,  aad  the  corrected 
anomaly  at  the  side,  take  out  the  equations  from  Table 
XXXVIII.  Also,  with  the  correct  anomaly,  take  out  the 
equation  from  Table  XXXIX.  With  the  sum  of  all  the  pre- 
ceding equations  at  top,  and  the  argument  of  variation  at  the 
side,  take  out  the  equation  from  Table  XL.  Also  with 
the  variation,  take  the  equation  from  Table  XLI.  With  the 
argument  of  reduction  take  out  the  equation  from  Table 
XLII.  These  equations,  all  added  together,  will  give  the 
true  hourly  motion  in  longitude. 

in  this  pro-  F°r  the  hourly  motion  in  latitude. —  With  the  1st  and  2d 
portion  the  arguments  of  latitude,  take  out  the  corresponding  quantities 
thTnaanmo*  ^rom  Cables  XLIII,  and  XLIV,  and  find  their  algebraic  sum, 
tion  of  the  noting  the  sign;  call  the  result  /. 

Then  make  the  following  proportion : 

32'  56"    :    L    :  :    I    :    ^ 

the  true  hourly  motion  in  latitude,  tending  north,  if  the  sign 
is  plus,  and  south,  if  minus.  In  this  proportion  L  is  the  true 
motion  of  the  moon  in  longitude,  and  the  first  term  is  the 
moon's  mean  motion ;  and  the  proportion  is  founded  on  the 
principle  that  the  true  motion  in  latitude  must  vary  by  the 
same  ratio  as  the  motion  in  longitude. 

N.  B.  In  computing  the  moon's  latitude  we  caution  the 
pupil  against  omitting  to  add  to  the  arguments  II,  V,  VI, 
VII,  VIII,  and  IX,  the  same  correction  as  to  the  column  of 
longitude ;  its  value  must  be  changed  into  the  decimal  division 
of  the  circle  for  all  the  columns  except  column  II. 

In  the  following  example  the  correction  for  longitude  is 
added  to  column  II,  and  its  value  to  all  the  following  columns 
except  column  X. 

We  find  the  value  in  question  thus : 

360°     :     13°  46'     :     :     1000     :    x. 

The  proportion  resolved  gives  ar  =  the  number  added  to 
the  several  columns. 

But  to  avoid  the  formality  of  resolving  a  proportion  for 
every  example,  we  g\ve  the  following  skeleton  of  a  table  that 


ECLIPSES.  281 

may  be  filled  out  to  any  extent  to  suit  the  convenience  and    CHAT.  m. 
taste  of  the  operator. 

Degrees  =  decimal  parti  Degrees    =    park. 

0  '  o      ' 

15     =     .003  5    24     =     .015 

1  26     =     .004  7     12     =     .020 

1  48     =     .205  90     =     .025 

2  10     =     .006  10    48     =     .030 
2  31     =     .007  12    36     =     .035 

2  53     =     .008  14    24     =     .040 

3  14     =     .009  16    12     =     .045 
3  36    =     .010 

To  make  use  of  this  table,  we  will  suppose  that  the  cor- 
rection for  longitude,  in  a  particular  example  is,  11°  31'  25  '; 
what  is  the  corresponding  decimal  or  numeral  part  ? 

Thus  9°         =     .025 

2  31     =         7 

11  31     =     .032 

We  now  continue  the  examples,  hoping  to  follow  these 
precepts. 


«82 


ASTRONOMY. 


CHAT.  Ill 


00  t-  <N  O  O   t- 
•^  CM  t~ 

CO  |  CO 


O5  05  CO  —  '  O 
<O  OS  00  CM 
O  CO  CM 


-.  CO  CO  CO  0 

—  i  co  co 

CO  •* 


- 

o 

CM 


O  O  CO  OS  O   t- 


co  >o  co  -<t  o 

OS  OJ  CO  CM 


t^  00  —  CO  ^ 
00  TJ«  OS  r- 
Tf  OS 


O  >O  CO  I-  O 
t^  00  r^  CM 

«3  »-i  CO 


fy.     00  "t  O  CO  ~*     •« 

J   -i  O  t-  «0     O 

cs  t—  r-     I  o 


00  •<*  OJ  O  O 
IO  CO  TT  CM 

OS  CO  CO 


CO  00  CM  CO  —  i 
O  OS  t-  O 
CO  CO  I-  # 


CO  t^  OS  O 

o;  co  ^ 
CO  CM 


t-  CO  CO  OS  ^t  i  CO 

r-  T  os  o    CM 

•^  CO  CM  CO    I  l- 

co  o  •*    !  oo 


o  —  i  co  co 

CO  OS  CO 
t^  CO  OJ 

O  CO 


CO  OS  — < 

o  — .  co 

rTco  CM~CO  -. 

•^  t-  -^  CM  -H 
CO  CM  Tj<  00 
CM_00  -• 

t^  Cs  Os  «C  t> 

— •  CO  — <  O 

Os  r-  r- 


!  S 


CO  Oi  CO  •<*  O 
—  '  O  O 

00  0 


TJ  CO  "^  00  O  00  iQO 
-JO  00  OS  CM  CO  l« 
0}  -^  CO  |<N 


t3  coco— 't^ooojir: 
HH  o  o  i>  CM  eo  !^r 
S  lo  -  «  o 


t-J       I' 

>_ 

> 


C5  1--  ^t  — '  O  OO    I O2 

o  -^  co  co      co  |o 

CO    -^   Tt1  iCb 


00  --D  rf  t^  O 
O  O  1—  O» 

CO  —1  CO 


O  C^  CO 

O    •* 


-^  00  CJ  00    CO 


co 

3 


g»o 

o 

J    w 


CO  CO  ^t  O    CO 


So 

w  . 


O  CO  "*  CM  -^  Tj" 

CO  -H  »0  CO         CO 

— i  t"  00  t-  t~  —• 

Tj«  r-,    CM    O  CO 


^  ^  ^  cr 

CM  OS  O    « 

*"*  ~*  *~*  <« 

.  ^»          ° 

CO 


§ 


o   I 

S 


.2        °  "5 


-a 

S 

o 


3 

JS 


x 
_0 

X3 


I 

tfl 

c 

'5 


i 


S 


S3 


o 

o 


ECLIPSES. 


283 


CHAP.  nL 


I  ^ 

-^j 


orsoj 


cc 


e  a  _ 

•2  -2  -    j> 


t'C 


;-2 
s  s 

b  5 


o  •- 

o 
o 


a 

*• 

H 

.2 

SS5 

^H 

"g 

1 

»ftiO 

(2 

C^ 

w 

10 

I0*~i 

3 

ji 

o 

i 

s 

Jq 

er 

a 

. 

'     B"  tAs?      .« 

1 

.2^.2     g 
HI    |Q 

wo  euoj 

00  C< 

"  -"t  co 


ift          O  r^ 


mm 


.84  ASTRONOMY 

CHAP.  III.  «       e      '        ' 

-     The  moon's  longitude,  as  just  computed,  will  be   9   20  15    9 
The  sun's  longitude,  at  the  same  time,  will  be     3   20  14  55 
The  difference  will  be  -  6     0    0  14. 

Therefore,  at  the  time  for  which  these  longitudes  were 
computed,  the  moon  will  be  past  her  full  by  14"  of  arc  :  to 
correct  the  time,  then,  we  must  find  how  much  time  will  be 
required  for  the  moon  to  gain  14"  ;  which,  by  the  problem 
of  the  couriers,  is 

14  _14^_        14^ 

-  (30.54)  —  (2.23)      28'  31"  ~~  1711* 
The  unit  for  t  is  one  hour,  and  the  denominator  of  the  frac- 
wbtractive     tion  is  the  difference  of  the  hourly  motions  of  the  sun  and 
the  rnoon,  as  determined  by  the  tables  ;  the  result  is  29  seconds 


The  Greenwich  time  will  be,  1851,  July  12d.  19h.  15m.  Os. 
would  be.i-      Subtract  -        _  29_ 

True  time  of  full  moon     -  12     19     14     31 

But  the  time  given  by  the  lunation  table  was  19  h.  14  m., 
differing  only  31  seconds  from  the  true  time  ;  the  approxi- 
mate and  true  time,  however,  do  not  commonly  coincide  as 
near  as  this  :  if  they  did,  none  but  the  most  rigid  astrono- 
mer would  use  the  lunar  tables  for  the  time  of  conjunction  or 
opposition. 

To  be  very  exact  we  must  correct  the  moon's  latitude  for 
what  it  will  vary  in  31  seconds  ;  that  is,  in  this  case,  increase 
it  1".5.  The  moon's  latitude,  at  the  time  of  full  moon,  is, 
therefore,  37'  13".4. 

We  have  now  all  the  elements  necessary  for  computing  the 
eclipse,  or,  at  least,  we  have  all  the  materials  for  finding 
them,  and,  for  convenience,  we  collect  the  elements  together  : 

d.         h.        m.        i. 

1.  True  time  of  full  moon,  July,     -     -     12     19     14     31 

2.  Semidiameter  of  earth's  shadow 

(page  265),  -      °    39'    39" 

3.  Angle  of  the  moon's  visible  path 

with  the  ecliptic,*        -         -  5    88     26 


Thif  's  the  angle  of  the  base  of  a  right-angled  triangle,  whoM  baw 


ECLIPSES.  285 

4.  Moon's  latitude  N.  descending,  37     13.4       

5.  Moon's  hourly  motion  from  the  sun,     -  28      31 

6.  Moon's  semidiameter,         -  15       4 

7.  Semidiameter  of  f)  and  earth's  shadow,  54     43 
Whenever  the  moon's  latitude,  at  the  time  of  full  moon,  is 

less  than  this  last  element,  the  moon  must  be  more  or  less 
eclipsed ;  and  it  is  by  computing  and  comparing  these  two  ele- 
ments, viz.,  4  and  7,  that  all  doubtful  cases  are  decided. 

TO    CONSTRUCT   A   LUNAR   ECLIPSE. 

From  any  convenient  scale  of  equal  parts,  take  the  7th  ele-  When  th« 
ment  in  your  dividers  (54  43)  =  54f ,  and  from  C,  as  a  center 
with  that  distance,  describe  the  semicircle  B  D  HE  (Fig.  55).  tjtude 
Take  CA  =  the  2d  element,  and  describe  the  semidiameter  scribe  a  ful1 
of  the  earth's  shadow.  From  C  the  center  of  the  shadow, 

\Vncn  lsrg$ 

draw  Cn  at  right  angles  to  B  E  the  ecliptic,  above  BE  when  g0nth  lati- 
the  latitude  is  north,  as  in  the  present  example,  but  below,  tude>  de' 

scribe      only 

if  south.  th.  lower 


Fig.  55. 


••micircl*. 


Take  the  moon's  latitude  from  the  scale  of  equal  parti, 
and  set  it  off  from  C  to  n.  Through  n  draw  Dnll,  the 
moon's  path,  so  that  the  line  shall  incline  to  B  E,  the  ecliptic, 
by  an  angle  equal  to  the  3d  element.  Conceive  the  moon's 

is  the  hourly  motion  of  the  moon  from  the  sun  (28' 31"),  and  the  per- 
pendicular, the  moon's  hourly  motion  in  latitude  (2' 49").  See 
page  266,  figure  54 


286  ASTRONOMY. 

JHAF  III    center  to  run  along  the  line  from  D  to  ff,  and  from  C  draw 
Cm  perpendicular  to  Dff. 

When  the  moon  is  ascending  in  her  orbit,  D  ZTmust  incline 
the  other  way,  and  Cm  must  lie  on  the  other  side  of  On. 

The  eclipse  commences  when  the  moon  arrives  at  D.     It  is 

the  time  of  full  moon  when  it  arrives  at  n  ;  the  greatest  ob- 

scuration occurs  when  it  arrives  at  m,  and  the  eclipse  ends  at 

ff.     The  duration  is  the  time  employed  in  passing  from  D  to 

H\  and  to  find  the  duration  apply  Dff  to  the  scale,  and  thus 

The  sth  eU-  find  its  measure.     Divide  this  measure  by  the  5th  element, 

ment  1S        and  we  shall  have  the  hours  and  decimal  parts  of  an  hour  in 

moon's  angu- 

lar    motion  the  duration.     Also  apply  Dn  to  the  scale  and  find  its  mea- 
from  the  sun  gure      Divide  this  measure  by  the  5th  element,  for  the  time 
of  describing  Dn,  also  divide  the  measure  nff  for  the  time  of 
describing  nff. 

The  time  of  describing  Dn,  subtracted  from  the  time  of 
full  moon,  will  give  the  time  of  the  beginning  of  the  eclipse; 
and  the  time  of  describing  nff,  added  to  the  time  of  full 
moon,  will  give  the  time  when  the  eclipse  ends. 

With  lunar  eclipses  the  time  of  greatest  obscuration  is  the 
instant  of  the  middle  of  the  eclipse,  provided  the  moon's  mo- 
tion from  the  sun,  for  this  short  period  of  time,  is  taken  as 
uniform,  as  it  may  be  without  sensible  error. 

In  reference  to  this  example  Dw  =  36'  and  nff  =44:'. 
These  distances,  divided  by  28'  31",  give  1  h.l4m.!6s.  for  the 
time  of  describing  Dn,  and  1  h.  32  m.  40  s.  for  nff:  whole 
time,  or  duration,  2  h.  27  m.  20  s. 

h.  m.  s. 

Astronomi- 

cal time  con        Therefore  from  the  time  of  full  C  19  14  31 


into      Subtract  -  -  -  1  14  16 

civil  time. 


Eclipse  begins  -             -  -  18    0  15 

Add  the  duration  -  -  2  47  20 

Eclipse  ends  -            '•-  -  20  47  35 

This  eclipse 


That  Is>  in  1851'  July  12d*  18  L  °  m'  15  8'    mean  astr°n°- 

wby.  mical  time,  the  eclipse  begins  ;  but  this  time  corresponds  with 

July  13,  at  6  h.  0  m.  in  the  morning;  and  at  this  time,  the  sun 
will  be  above  the  horizon  of  Greenwich,  and,  of  course,  the 


ECLIPSES.  287 

full  moon,  which  is  always  opposite  to  the  sun,  will  be  below    CHAP.  ra. 
the  horizon,  and  the  eclipse  will  be  invisible  to  all  Europe.     vi*iti«  fa 

In  the  United  States,  however,  the  eclipse  will  be  visible;  the  u.s 
for,  at  these  points  of  absolute  time,  the  sun  will  not  have 
risen  nor  the  moon  have  gone  down  ;  but,  to  be  more  definite, 
we  demand  the  times  of  the  beginning,  middle,  and  end  of  the 
eclipse,  as  seen  from  Albany,  N.  Y.  To  answer  this  demand, 
all  we  have  to  do,  is  to  subtract  from  the  Greenwich  time  the 
difference  of  meridians  between  the  two  places,  which,  in  this 
case,  is  4  h.  55  m.  ;  and  the  result  is, 

Beginning  of  the  eclipse  13  d.  1  h.    5m.  morning, 

Middle  2      30 

End  of  the  eclipse  3      52  „ 

In  the  same  manner  we  would  compute  the  time  for  any 
other  place. 

For  the  quantity  of  the  eclipse  we  take  the  portion  of  The  qnan- 
the  moon's  diameter,  which  is  immersed  in  the  shadow,  tlty  of  the 

eclipse    how 

at    the  time  of  greatest   obscuration,  and    compare  it  with  found. 
the  whole  diameter  of  the   moon;    and  in  the  present   ex- 
ample, we  perceive,  that  more  than  half  of  the  diameter  is 
eclipsed  —  about  7  digits  when  the  whole  is  called  12,  or  0.6 
when  the  diameter  is  1. 

All  these  results,  however,  except  the  time  of  full  moon, 
are  approximate,  because  we  cannot,  nor  do  we  pretend  to 
construct  to  accuracy  ;  but  any  mathematician  can  obtain  accurate 
results  by  means  of  the  triangles  D  C  H  and  C  nm,  and  the 
relative  motion  of  the  moon  from  the  sun. 

In  the  right-angled  triangle  Cnm,  right-angled  at  m,  Cn  The  exact 
is  the  latitude  of  the  moon  =  37'  17".4  =  2237".4,  and  the  computation 
angle  n  Cm  =  5°  38'  26"  ;  with  these  data  we  find  m  n  =  tion  Ol  ^ 


,  and  Cm  =  2212"  •clip". 

In  the  right-angled  triangle  C  Dm,  or  its  equal  CmH,  we 
Uve  - 

Or,        -         - 

Or,       -         -    mH2=(Cff+Cm)  (Off—  Cm). 
Cffis  the  7th  element  =  3283",  and  Cm  =  2212  '.6. 
Therefore,  m  ff=  7(5495)    (1071  )  =  2426"         This 


288  ASTRONOMY. 


"'•  divided  by  1711",  the  5th  element,  gives  the  time  of  half 
the  duration  of  the  eclipse  Ih.  25m.;  therefore  the  whole  du- 
ration is  2  h.  50m.,  which  is  2  m.  40  s.morethan  the  time  we 
obtained  by  the  rough  construction. 

The  distance  nm,  as  just  determined,  is  220",  and  the  time 
of  describing  this  space,  at  the  rate  of  1711"  per  hour,  re- 
quires  7  ra.  52  s.,  which  taken  from  and  added  to  the  semi- 
duration,  gives  1  h.lTm.  8  s.  from  the  beginning  of  the  eclipse 
to  full  moon,  and  1  h.  32  m.  52  s.  from  the  full  moon  to  the 
end  of  the  eclipse. 

The  uigo-      por  the  magnitude  of  the  eclipse,  we  add  the  moon's  semi- 
diameter  in  seconds  (904"  )  to  Cm  (  2212"  ),  and  from  the 


ofthemagni-  sura  subtract  the  semidiameter  of  the  shadow  in  seconds 
f  he  (  2379  ),  and  the  remainder  is  the  portion  of  the  moon's  di- 
ameter not  eclipsed.  Subtract  this  quantity  from  the  moon's 
diameter,  and  we  shall  have  the  part  eclipsed.  Divide  this 
by  the  whole  diameter,  and  the  quotient  is  the  magnitude  of 
the  eclipse,  the  moon's  diameter  being  unity. 

Following  these  directions,  we  find  the  magnitude  of  this 
eclipse  must  be  0.587. 

The  con.  jn  ajj  these  computations  we  were  guided  by  the  construc- 
sufficient  tion  ;  which  uill  always  prove  a  sufficient  index,  and  all  that 
guide  to  car.  should  he  required. 

uigonometrit      ^e  mav  determine,  in  any  case,  whether  the  eclipse  will  or 
cai  computa-  will  not  be  total,  by  the  following  operation  : 

Subtract  the  $)'s  semidiameter  from  the  semidiameter  of 
the  shadow,  and  if  the  moon's  latitude,  at  the  time  of  full 
moon,  is  less  than  the  remainder,  the  eclipse  will  be  total, 
otherwise  not. 

To  find  the  duration  of  total  darkness.  —  Diminish  the  semi- 
diameter  of  the  shadow  by  the  semidiameter  of  the  moon,  and 
from  the  center  of  the  shadow  describe  a  circle,  with  a  radius 
equal  to  the  remainder;  a  portion  of  the  moon's  path  must 
come  within  this  circle  ;  that  portion,  measured  or  divided  by 
the  hourly  motion,  will  give  the  time  of  total  darkness. 

When  the  moon's  latitude  is  north,  as  in  the  present  ex- 
ample, the  southern  limb  of  the  moon  is  eclipsed  —  and  con- 
verielv 


ECLIPSES 


CHAPTER   IV. 

SOLAR    ECLIPSES GENERAL    AND    LOCAL. 

THE  elements  for  a  solar  eclipse  are  computed  in  the  same  CHAP.  iv. 
manner  as  the  elements  of  a  lunar  eclipse ;  all  of  which  are  General  di 
found  by  the  solar  and  lunar  tables.  rections  tc 

_.        J  find   the  ele- 

The  approximate  time  of  new  moon  is  first  computed,  and  mentg 
for  this  time,  compute  the  sun's  longitude,  declination,  paral- 
lax,  semidiameter,   and    hourly   motion;    and   for   the  same 
time  compute  the  moon's  longitude,  latitude,  hourly  motion  in 
longitude  and  latitude,  horizontal  parallax,  and  semidiameter. 

If  the  longitudes  of  both  sun  and  moon  are  found  to  be  the 
same,  then  the  approximate  time  of  conjunction;  found  by  the 
lunation  tables,  is  the  same  as  the  true  time ;  if  not,  we  pro- 
portion to  the  true  time,  as  described  in  the  last  chapter. 

The  elements  for  a  general  solar  eclipse  are : 

1.  The  time  of  ^  *  ai  some  known  meridian.  2.  Longi-  what  ele. 
tude  of  O  and  f).  3.  Q's  declination.  4.  f)'s  latitude.  ments  ara 

w  v-/  necessary. 

5.  O's  hourly  motion.  6.  O's  hourly  motion  in  longitude. 
7.  O's  hourly  motion  in  latitude.  8.  The  angle  of  the  O's 
visible  path  with  the  ecliptic.  9.  O's  horizontal  parallax. 
10.  f)'s  semidiaraeter.  11.  Q's  semidiameter.  12.  Q's 
horizontal  parallax. 

For  a  local  eclipse,  the  latitude  of  the  particular  locality 
must  also  be  given,  or  considered  as  one  of  the  elements. 

As  we  can  best  illustrate  general  principles  by  taking  a    A  definite 
particular  example,  we  now  propose  to  show  the  general  course  exal"p  e 
yf  an  eclipse  of  the  sun,  which  will  occur  in  May  1854;  where 
it  will  first  commence  on  the  earth ;  in  what  latitude  and  longi- 
tude the  sun  will  be  centrally  eclipsed  at  noon,  and  where ;  in 
what  latitude  and  longitude  the  eclipse  will  finally  leave  the  earth. 

We  speak  of  an  eclipse  of  the  sun  being  on  the  earth ;  by  Some  ?ene- 
this  we  mean  the  moon's  shadow  on  the  earth.     If  an  observer  ™ 
is  in  the  moon's  shadow,  of  course,  the  sun  would  be  in  an  natiow 
eclipse  to  him ;  and,  if  a  tangent  line  be  drawn  l«tween  the 

•  Sign  of  conjunction 
19 


S90 


ASTRONOMY 


CHAP.  nr.  BUn  and  moon,  and  that  line  strike  the  eye  of  an  ohserver  on 
the  earth,  to  that  observer  the  limbs  of  the  sun  and  iLoon 
would  apparently  meet,  and  all  projections  of  eclipses  are  on 
the  principle  of  lines  drawn  from  some  part  of  the  sun  to 
some  part  of  the  moon,  and  those  lines  striking  the  earth. 
When  no  such  lines  can  strike  the  earth  there  can  be  no 
eclipse.  For  the  sake  of  simplicity  in  explaining  a  projection 

Point  of    °f  a  s°lar  eclipse,  whether  it  be  general  or  local,  an  observer 

i«*  is  supposed  to  be  at  the  moon,  looking  down  on  the  earth, 

viewing  the  moon's  shadow  as  it  passes  over  the  earth's  disc; 

and,  of  course,  the  earth  to  him  appears  as  a  plane,  equal  to 

the  moon's  horizontal  parallax. 

The  approximate  time  of  new  moon  will  be  found  com- 
puted on  page  254,  and,  if  very  close  results  are  not  required, 
we  may  compute  the  sun's  longitude,  declination,  hourly  mo- 
tion, and  semidiameter  for  this  time,  and  take  out  the  moon's 
-  horizontal  parallax,  hourly  motion,  and  semidiameter  from 
Table  XIV;  but  we  have  computed  the  elements  more  accu- 
rately by  the  lunar  tables,  and  find  them  as  follows : 

d.       h.     m.      ». 

1.  Greenwich  mean  time  of  ^   1854,  May  26    8  45  39 

2.  Lon.  of  O  and  f)         -        -         -          65°  14'   6" 

3.  Declination  of  the  Q             -         -            21  11  43  K 

4.  Latitude  of  the  O                          -  21  19  N. 

5.  O's  hourly  motion  in  Ion.,     -         -         -  2  24 

6.  C>'s  hourly  motion  in  Ion.,     -         -  30     3 

7.  f)'s  hourly  motion  in  lat.,  tending  north,  2  46 
From  5,  6,  and  7  we  obtain  8,  as  explained 

in  the  last  chapter. 

8.  Angle  of  the  moon's  visible  path  o     '      " 

with  the  eclip.,  -        -     5  42   50 

9.  The  f)'s  horizontal  equatorial  parallax,  54   30 

10.  The  f)'s  semidiameter,  -  -         14   51 

11.  The  O's  semidiameter,  -  -        15   48 

12.  The  O's  horizontal  parallax,  always  taken  at  9 
Subtract  the  O  's  horizontal  parallax  from  the  £) 's  ;  and 

to  the  remainder  the  semidiameters  of   O   aQd  €) ,  and  if 
the  moon's  latitude  is  less  than  this  sum,  there  will  be  an 


Accurate 
element*   for 
the  solar 
eclipse, 
which    '  will 
take      place 
Maj     28, 
18S4. 


ECLIPSES.  291 

eclipse,  otherwise  not;  and  it  is  by  comparing  this  sum  with  CHAP.IV. 
the  moon's  latitude  that  all  doubtful  cases  are  decided 

TO   CONSTRUCT    A    GENERAL    ECLIPSE. 

1.  Make,  or  procure,  a  convenient  scale  of  equal  parts,  and 
from  any  point  as  C  ( Fig.  56  )  with  the  radius  CJB,  equal  to 
the  difference     of  the    parallaxes  of  Q  and  O  (in  the  pre- 
sent example  54' 21",  the  minute  is  the  unit),  describe  tho 
semicircle  C  B  P  H,  or  the  whole  circle,  when  the  case  re- 
quires it.     When  the  moon  has  small  latitude  (less  than  20') 
describe  the  whole  circle ;  when  the  moon  has  large  north  lati- 
tude, describe  the  northern  semicircle;  when  south,  describe  the 
southern  semicircle. 

Through  C  draw  VCD  PL  perpendicular  to  HE.  This 
perpendicular  will  represent  the  plane  of  the  earth's  axis,  as 
seen  from  the  moon. 

From  P  take  P  A,  PF,  each  equal  to  the  obliquity  of  the 
ecliptic  23°  27'  30",  and  draw  the  chord  A  F. 

On  A  F,  as  a  diameter,  describe  the  semicircle  ALF.          ^  the  M 

2.  Find  the  distance  of  the  sun  from  the  tropic,  nearest  to  of  ihe 
it,  by  taking  the  difference  between  the  sun's  longitude  and  Uo* 
90°  or  270°,  as  the  case  may  be.     In  the  present  example  we 
subtract  65°  14'  from  90°,  the  remainder  is  24°  46'.     Take 

L  T,  equal  to  24°  46',  and  draw  TE  parallel  to  L  C.     Draw 
C  E  the  axis  of  the  ecliptic. 

By  the  revolution  of  the  earth  round  the  sun,  the  axis  of      T1* 
the  ecliptic  appears  to  coincide  with  the  axis  of  the  equator,  °ic    ° 
when  the  sun  is  at  either  tropic,  and  it  appears  to  depart  iu  p<mtioo. 
from  that  line  by  the  whole  amount  of  the  obliquity  of  tho 
ecliptic ;  and  the  time  of  this  greatest  departure  is  when  the 
sun  is  on  the  equator.     That  is,  CE  runs  out  to  C  A  at  the 
vernal  equinox,  and  runs  out  to  C  F  at  the  autumnal  equi- 
nox.    As  a  general  rule,  CE,  the  axis  of  the  ecliptic,  is  to 
the  left  of  OP,  the  axis  of  the  equator,  from  the  20th  of  De- 
cember to  the  20th  of  June,  and  to  the  right  of  that  line  the 
rest  of  the  year.     Draw  C  0  the  axis  of  the  moon's  orbit,  so  a°*  *  **» 
that  the  angle    O  CE  shall  be  equal  to  the  angle  of  the  the  IttBM  a 
moon's  visible  path  with  the  ecliptic,  and  CO  is  to  the  left  of  w*. 


292  ASTRONOMY. 

CHAP.  iv.  C  E  when  the  eclipse  is  about  the  ascending  node,  as  in  this 
example,  but  at  the  right  when  the  eclipse  is  about  the  de- 
cending  node. 

For  this  projection  to  appear  natural,  the  reader  should 
face  the  north,  so  that  H  will  appear  to  the  west,  and  B  on 
the  east  of  the  figure. 

The  shadow  of  the  moon  across  the  earth  is  from  a  western 
to  an  eastern  direction,  therefore,  the  moon  is  conceived  to 
come  in  on  the  earth  from  the  west  side. 

The  eqna-  rpjic  ^Q^  (j  jg  perpen(Jicular  to  the  sun's  declination,  and 
C  V  is  the  sine  of  the  declination,  and  the  curved  line  H  VB 
is  a  representation  of  the  equator  as  seen  from  the  moon. 
When  the  sun  has  no  declination,  the  equator  draws  up  into 
a  straight  line. 

HOW     to      3.  Take  C  n  from  the  scale  of  equal  parts,  making  it  equal 
lrawt     the  to  the  moon's  latitude,  and  through  the  point  n,  and  at  right 
angles  to  C  G,  draw  the  line  klmnrpq,  which  represents  the 
center  of  the  shadow,  or  the  moon's  path  across  the  disc. 

From  C  as  a  center,  at  the  distance  C  0,  describe  the 
outer  semicircle,  equal  to  the  diff.  of  the  moon's  horizontal 
parallax,  the  sun's  horizontal  parallax,  and  the  semidiameter 
of  both  sun  and  moon  ;  then  0  H  is  the  semidiameter  of  the 
sun  and  moon. 

When  the  eclipse  first  commences,  the  center  of  the  moon 
is  at  k,  and  the  center  of  the  sun  is  on  the  circumference  of 
the  other  circle,  in  a  direct  line  to  (7,*hot  represented  in  the 
figure,  therefore,  the  two  limbs  must  then  just  touch. 

As  C  is  the  center  of  the  earth,  and  H  on  the  equator, 
therefore  CH  0  is  a  line  in  the  plane  of  the  equator,  and  the 
point  k  is  a  little  below  the  equator  ;  which  shows  that  the 
eclipse  first  commences  on  the  earth  a  little  south  of  the 
equator. 

MOW  to  o>-      The  time  that  the  eclipse  is  on  the  earth  is  measured  by 
ie  t"ne  rccluired  f°r  ^e  moon  to  PasB  fr°m  £  to  7 


fwurfti         true  angular  motion  from  the  sun. 

•clime.  rpke  iength  of  this  line,  k  q,  can  be  found  from  the  ele- 

ments, and  trigonometry,  as  in  an  eclipse  of  the  moon,  and 
the  tirae  of  describing  it  is  found  in  the  same  way. 


ECLIPSES. 


298 


294  ASTRONOMY. 

OH  A*,  iv.       When  the  moon's  center  comes  to  I,  the  central  eclipse 
HOW  u>  de-  commences,  and  the  arc  HI  shows  that  it  must  be  about  in 
^mine     m  the  latitude   of  7°  north.     When  the  moon's  center  comes 
t-i.ies      the  to  **>  the  sun  will  be  centrally  eclipsed  at  apparent  noon ;  and 
eclipse    will  Cr  is  the  sine  of  the  number  of  degrees  north  of  the  sun's 
over,'     and  declination,  which,  in  this  case,  is  about  23° ;  hence  to  the 
pass  off  the  sun's  declination,  21°  12',  add  23°,  making  44°  12';  showing, 
as  near  as  a  mere  projection  can  show,  that  the  sun  will  be 
centrally  eclipsed  at  noon  on  some  meridian,  in  latitude  44°  12 
north.     The  central  eclipse  will  end,  or  pass  off  the  earth, 
when  the  moon's  center  arrives  at  p  and  the  arc  Ep  from  the 
equator,  shows  that  the  latitude  must  be  about  41°  north.    The 
eclipse  will  entirely  leave  the  earth  when  the  moon's  center 
arrives  at  q,  arid  for  its  limb  to  touch  the  sun,  the  sun's  cen- 
ter must  be  at  h,  and  the  arc  E  h  shows  that  the  latitude 
must  be  about  30°  north. 

The  lines,  cd  and  ab,  parallel  to  the  moon's  path,  and  dis- 
tant from  it  equal  to  the  sum  of  the  semidiameters  of  sun  and 
moon,  represent  the  lines  of  simple  contacts  across  the  earth, 
or  limits  of  the  eclipse ;  cd  is  the  southern  line  of  simple  con- 
tact, and  a  b  is  the  northern  line  of  simple  contact,  and  the 
latitudes  at  which  these  lines  make  their  transits  over  the 
earth,  are  determined  precisely  as  the  latitudes  on  the  cen- 
tral line. 

We    may      J$\it  we  need  not  stop  at  coarse  approximations:  we  have 
rate  compn-  all  the   data  for  correct  mathematical  results,  on  the  same 
*  tations     by  principles  as  we  determined  those  in  relation  to  a  lunar  eclipse. 
«,"*         In  the  triangle    Cnr,  we  have  the  side    Cn,  the  moon's 
latitude  in  secunds,  which  may  be  used  as  linear  measure,  as 
yards  or  feet  and  in  proportion  thereto,  we  may  compute  Cr 
and  nr,  when  we  know  tJie  anyle  n  Cr. 

An  eqoa-  jjut  ^  f0l}owj,,g.  equation  always  gives  the  tangent  of  the 
position  of  angle  E  CD  or  n  Cr,  calling  the  sun's  distance  from  the  sol- 
the  axu  of  gtice  D,  the  obliquity  of  the  ecliptic  E,  and  the  radius  unity. 

tlm  ecliptic. 

tan.  J£C2)=tan.  E  sin.  D.* 


*  The  student  who  has  acquired  a  little  skill  in  analytical  tri|?»o- 
metry  can  discover  the  preliminary  steps  to  this  equation;  the  princi- 
ples are  all  visible  in  the  construction  of  the  figure. 


ECLIPSES  296 

To  the  angle  E  CD,  add  the  angle   Q  CJS,  the  angle  of  the    CHAF  nr. 
moon's  visible  path  with  the  ecliptic,  and  we  have  the  whole 
angle  G  C D,  or  m  Or.     Cmn  is  a  right  angle;  and  in  the 
two  triangles    Cm  n  and  Cmr,  we  have  all  the  data,  and  can 
compute  n  r  and  r  C. 

When  the  moon  arrives  at  mt  it  is  in  the  line  of  conjunction 
in  her  orbit ;  when  it  arrives  at  »,  it  is  in  ecliptic  conjunction ; 
and  when  it  arrives  at  r,  it  attains  conjunction  in  right  as- 
cension. 

For  the  last  six  or  eight  years,  the  English  Nautical  Al-         Recent 

i  .  .1  ...  -,  .  .  •        «    i  ,  changes      in 

manac  has  given  the  conjunctions  and  oppositions  in  right  as-  the   English 
eension,  in  place  of  conjunctions  and  oppositions  in  longitude,  Nautical  Al- 
and has  given  the  difference  of  declinations  between  the  sun  m< 
and  moon,  in  place  of  giving  the  moon's  latitude ;  that  is,  it 
has  given  the  time  that  the  moon  arrives  at  r,  in  place  of  n, 
and  given  the  line  Cr  in  place  of  Cn. 

All  lunar  tables  give  the  ecliptic  conjunction  at  n,  and  from 
this  we  can  compute  the  time  at  r  by  means  of  the  triangle 
Cnr. 

Having  explained  the  principle  of  finding  the  latitude  on 
the  earth,  when  a  solar  eclipse  first  commences,  we  are  now 
ready  to  show  another  important  principle — how  to  find  the 
longitude ;  and  with  the  latitude  and  longitude,  we  have  the 
exact  point  on  the  earth. 

Where  an  eclipse  first  commences  on  the   earth,  it  com-  The  method 
mences  with  the  rising  sun,  and  finally  leaves  the  earth  with  "Jjf "f^JI U 
the  setting  sun.     In  this  example,  we  have  decided  that  the  where     the 
eclipse  must  commence  very  near  the  equator,  not  more  than  *££**    fi^ 
one  degree  south;  but  in  that  latitude  the  sun  rises  at  6  h.  earth. 
A.  M.  apparent  time ;  therefore,  at  the  place  where  the  eclipse 
commences,  it  is  six  in  the  morning,  apparent  time. 

From  the  scale  of  equal  parts,  take  the  moon's  hourly  mo- 
tion from  the  sun  in  the  dividers  (27'  39"),  and  apply  it  on 
the  line  k  q:  it  will  extend  three  times,  and  a  little  over,  to  the 
point  ».  This  shows  that  three  hours,  and  a  little  more  ( we 
say  3h.  3m.)  must  elapse  from  the  first  commencement  of 
the  eclipse  to  the  change  of  the  moon  at  n.  Hence,  by  the 
local  time  at  the  place  of  the  commencement  of  the  eclipse, 


295  ASTRONOMY. 

CHAP,  iv.  the  moon  changes  at  9  h.  3  m.  in  the  morning,  apparent  time ; 
but  the  apparent  time  of  new  moon  at  Greenwich  is  8  h.  49  m. 
p.  M.,  making  a  difference  of  11  h.  46m.  for  mere  locality: 
the  absolute  instant  is  the  same;  the  difference  is  only  in 
meridians  which  correspond  to  a  difference  of  longitude  of 
175°  30';  and  it  is  west,  because  it  is  later  in  the  day  at 
Greenwich. 

of  'fiadiM  ^e  central  eclipse  also  first  comes  on  the  earth  at  a  place 
where  tbe  where  the  sun  is  rising.  In  this  example  it  first  strikes  the 
ce"trae1  ^  earth  at  the  point  /,  in  latitude  about  7°  N. ;  but,  in  latitude 
strikes  the  ^°  N.,  and  declination  21°  N.,  the  sun  rises  at  5h.  48m., 
earlh  A.  M.  apparent  time  (  Prob.  II );  and  from  that  time  to  the 

change  of  the  moon,  namely,  the  time  required  for  the  moon 
to  move  from  /  to  n,  is  (  as  near  as  we  can  estimate  it  by  the 
construction  ),  1  h.  56  m.;  therefore,  the  time  of  new  moon,  in 
the  locality  where  the  central  eclipse  first  commences,  is  7  b. 
44  m.  in  the  morning.  From  this  to  8  h.  49  m.  in  the  even- 
ing, the  time  at  Greenwich,  gives  a  difference  of  13  h.  5m, 
reckoned  eastward  from  the  locality,  or  10 h.  55m.  reckoned 
westward ;  which  corresponds  to  196°  15'  west  longitude  from 
Greenwich,  or  163°  45'  east  longitude;  the  meridian  is  the 
same.  If  the  longitude  is  called  east,  the  day  of  the  month 
must  be  one  later;  but,  to  avoid  this,  wo  had  better  call  the 
lungitude  west. 
To  find  the  Where  the  sun  is  centrally  eclipsed  on  the  meridian,  it  is 

whfre"  8the  Jus^  ^~'  aPParenfc  ^me  >  *ne  moon's  center  is  then  at  r,  and, 

snn  will  be  by  the  construction,  it  must  be  about  seven  minutes  after 

»ed7  at  GonJunct*on  *n  ^at  locality ;  hence,  the  conjunction  is  seven 

noon  minutes  before  12,  and  at  Greenwich  it  is  8  h  49  m.  after  12, 

giving  8  h.  56  m.  for  difference  of  longitude,  or  134°  west 

longitude 

The  central  eclipse  will  leave  the  earth  with  the  setting 
sun,  when  the  center  of  the  moon  and  sun  are  both  at  p;  but 
the  latitude  of  p  we  decided  to  be  40°  north,  and  in  this 
latitude,  when  the  sun's  declination  is  21°  11',  as  it  now 
is,  the  sun  sets  at  7h.  15m.  apparent  time;  but  this  is 
1  h.  40  m.  after  conjunction,  therefore  the  conjunction  in 
that  locality  must  be  at  5  h.  35  m. ;  but,  at  Greenwich,  it  if 


ECLIPSES  297 


8h.  49m.,  giving,  for  difference  of  longitude,  3h.  14m.,  or 
48°  30'  west. 

The  eclipse  finally  leaves  the  earth  in  latitude  46°  north  ;    To  find  th« 
but,  in  this  latitude,  the  sun  sets  at  6  h.  51  m..  and  the  con-  lo°?ltnde 

where        tn® 

junction  will  be  3  h.  0  m.  sooner  (  the  time  required  for  the  eciip»e   will 
moon  to  pass  from  n  to  ^),  therefore  the  conjunction  in  this 
locality  must  be  at  3h.  51m.;  but,  at  Greenwich,  it  will  be 
8h.  49m.,  giving  4h.  58m.  for  difference  of  longitude,  or 
74°  30'  west. 

Thus,  by  the  mere  geometrical  construction,  we  have 
roughly  determined  the  following  important  particulars  : 

App.  time  Gr.  Lat.  Longitude. 

h.    m.  °  o         / 

Eclipse  commences,  May  26,    5  46          IS.  175  30  W. 

Cen.  eclipse  commences,  6  53          7  N.  196  15  W. 

Cen.  eclipse  at  local  noon,          8  56  46  134  00  W.  the    P«>jec. 

Cen.  eclipse  ends,  10  34  40  48  30  W.  tlon< 

End  of  eclipse,  11  46  30  73  30  W. 

To  find  the  latitude  of  the  first  commencement  of  simple    The  locali- 
contact  on  the  southern  line,  all  we  have  to  do  is  to  find  the  southern  and 
arc  #c;and  for  the  latitude  on  the  northern  line,  we  find  the  northern 
arc  Ha  ;  the  point  c  is  in  latitude  about  27°  south,  and  a  in  p]"e*0°ta*!*" 
about  54°  north. 

The  southern  line  of  simple  contact  leaves  the  earth  at  d, 
between  the  seventh  and  eighth  degrees  of  north  latitude,  and 
the  northern  line  passes  off  beyond  the  pole. 

We  have,  thus  far,  taken  the  results  but  approximately 
from  the  projection,  and  the  projection  is  sufficient  to  teach 
us  principles  ;  and  it  must  be  our  guide,  if  we  attempt  to  ob- 
tain more  minute  results  ;  and  with  the  elements  and  the  figure 
we  have  the  whole  subject  before  us  as  minutely  accurate 
as  it  is  magnificent,  and  as  simple  as  it  is  sublime. 

To  complete  our  illustration,  we  now  go  through  the  trigo- 
nometrical computation. 

In  the  triangle  Cnm,  we  have  C7w=21'  19"=1279,  the 
angle  mCn=5°  42'  50",  and  the  angle  m  a  right  angle. 

Whence  Cro=1273",  and  m»=127".3. 


298  ASTRONOMY. 

CHAP,  iv.    tan.  E  CD=n  CV=tan.  (23°  27'  32")  sin.  (24°  45'  54") 

In     these  (  page  284). 

ti°ons2"          Whence  E  CD=  10°  18'    8", 

moon>8  lati-      Add  G  C  E=  5°  42'  50", 

tude  and  the  -  •  •  --  -  • 

distances          Sum  is      Q  C  D=m  CV=16°    0'  58". 

In  the  trianSle  m  Cr>  we  have  Om  (1273),  the  perpendieu- 
lar,  and  the  angle  m  Cr  as  just  determined  ;  whence, 

mr=365".3  ;       CV=1324".3. 

In  the  triangle    Cmp,  Cp  is  the  horizontal  parallax  of 
moon  and  sun  (54'  30")—  9",  or  54'  21"=3260". 

By  the  well-known  property  of  the  right-angled  triangle, 


Or     mp2  =  Cp2—  Cm2=(Cp+Cm)  (Cp—Cm), 

That  is,      wjp=>/(4533)(1987)=3001/'.7. 

Therefore,  Ip,  the  whole  chord,  is  6003".4,  which,  divided 
by  1659'  (the  moon's  motion  from  the  sun),  gives  3.616h. 
or  3  h.  37m.  40s.  for  the  time  that  the  central  eclipse  will 
be  on  the  earth. 

In  the  same  manner  the  line  m  q  is  found. 

That  is,        mq=J(Cq-\-Cm)(Cq—  (7m), 

But,         C?=54'  21"-fl4'  5.1'H-15'  48"=5100". 


Or  m  q=  >/(6373)(3827)=4938".3. 

Therefore,  the  whole  chord,  kq,  is  9876.6,  which,  divided  by 
1659",  gives  5  h.  57  m.  20  s.  for  the  entire  duration  of  the 
general  eclipse  on  the  earth. 

On  the  supposition  that  the  moon's  motion  from  the  sun  is 
uniform  for  the  six  hours  that  the  eclipse  will  be  on  the  earth, 
the  several  parts  of  the  moon's  path  will  be  passed  over  by 
the  moon,  as  follows  : 

From    *to*in111-    9m.  54s. 
condition  of     From    /  to  m  in  1      49       00    to  <J  in  orbit. 
invariable  el-      From  m  to  n  in  4       36     to  c$  in  ecliptic. 

From   n  to  r  in  8       37     to  £  in  right  ascension. 

From    rtojt?  in  1      35      40 

From   p  to  q  in  1        9       54 


ECLIPSES.  299 

The  apparent  time  of  ecliptic  conjunction,  at  Greenwich,  cm*.  IT. 
as  determined  by  the  tables  (  and  applying  the  equation  of 
time),  is  at  8  h.  49  m.    0  s. 

Subtract  from  k  to  ecliptic  <*> ,  3        3       30 

Eclipse  commences,  Greenwich  app.  time,        5      45       30 
Central  eclipse  commences  (add  1  9  54),         6      55       24 
Sun  centrally  eclipsed  on  some  meridian,  or 
d  in  right   ascension,    Greenwich   time, 

at  (add  2  2  36),  8      58      00 

Central  eclipse  ends  at  (add  1  35  48),  10      33       48 

End  of  eclipse  at  (add  1  9  54),  11      43       41 

By  comparing  these  times  with  those  obtained  simply  by 
the  projection,  we  perceive  that  the  projection  is  not  far  out 
of  the  way,  notwithstanding  the  terms  rough  and  rougJdy  that  rate>  than 
we  have  been  compelled  to  use  concerning  it.     Indeed,  a  good  generally 
draftsman,  with  a  delicate  scale  and  good  dividers,  can  decide  inpl>08< 
the  times  within  two  minutes,  and  the  latitudes  and  longitudes 
within  half  a  degree ;  but  all  mathematical  minds,  of  course, 
prefer  more  accurate  results;    yet,  however  great  the  care, 
absolute  accuracy  cannot  be  attained ;   the  nature  of  the  case 
does  not  admit  of  it.* 

To  find  whether  the  point  k  is  north  or  south  of  the  equa- 

*The  astronomer,  by  making  use  of  his  judgment,  can  be  very  ac- 
curate with  very  little  trouble:  he  perceives,  at  a  glance,  what  ele- 
ments vary,  and  what  the  effects  of  such  variation  will  be;  but  a  learner, 
who  is  supposed  not  to  be  able  to  take  a  comprehensive  view  of  the 
whole  subject,  must  go  through  the  tedious  process  of  computing  the 
elements  for  the  times  of  the  beginning  and  end  of  the  eclipse,  as  well 
as  the  time  of  conjunction,  if  he  aims  at  accuracy,  but  an  astronomer 
can  be  at  once  brief  and  accurate.  In  computing  the  moon's  longi- 
tude, in  the  present  example,  the  astronomer  would  notice  in  particu- 
lar the  moon's  anomaly,  and,  by  it,  he  perceives  whether  the  moon's 
hourly  motion  is  on  the  increase  or  decrease,  and  at  what  rate. 

It  is  on  the  decrease, and  the  first  part  of  the  chord  km  is  passed  over 
by  the  moon  in  about  7  seconds  less  time  than  our  computation 
made  it,  and  the  last  part  requires  about  7  seconds  longer  time  ;  but 
the  times  of  passing  m  and  n  should  be  considered  accurate,  and  the 
times  of  beginning  and  end  should  be  modified  for  the  variation  of 
the  moon's  motion,  making  the  beginning  and  end  7  seconds  later,  and 
the  beginning  and  end  of  the  central  eclipse  about  4  seconds  later. 


300  ASTRONOMY. 

.?jy.  tor,  we  conceive  k  and  C  joined,  and  if  the  angle  m  Ck  is 
greater  than  the  angle  m  Off,  the  point  k  is  south,  otherwise 
north. 

By  trigonometry,  Ck   :  km  :  :  sine  90°  :  sine  mCk; 

o       /        // 

Or,       5138  :  4900".3  :  sin.  90  :  sin.w  Cfe75  31  20 
To  this  add  G  CD,        -  -  16    0  58 

Sum  is  the  angle  r  Ck  -  91  32  18 

This  angle  shows  that  the  eclipse  will  first  touch  the  earth 

in  latitude  1°  32'  18"  south. 

To  find  the  arc  HI,  conceive  the  points  Cl  joined,  and  the 

two  triangles  Clm,  m  C p  are  equal. 

Cl  :  Im  :  :  sin.  90°  :  m  Cl' 


Or,        3261  :  3003.7  :  :  sin.  90  :  sin.  m  C7=67  7  50 
To  this  add  G  C  £,  v          160  58 

The  sum  is,  83  8  48 

Where  the  This  angle  shows  the  latitude  of  the  point  /  to  be  6°  51' 
•triEa  iht"*  *2"  norfcn-  That  is»  *be  central  eclipse  first  touches  the 
earth  earth  in  6°  51'  12"  of  north  latitude;  differing  very  little  from 

the  point  determined  by  construction. 

To  find  the  latitude  of  the  point  p,  we  have  m  Cl  =  m  Cp 
=  67°  7'  50";  and  subtracting  16°  0'  58",  we  have  the 
polar  distance,  or  co -latitude;  the  result  is,  that  the  central 
eclipse  passes  off  at  latitude  38°  53'  8"  north,  and  the  gene- 
ral eclipse  entirely  leaves  the  earth  in  latitude  30°  25'  38". 

To  find  the  latitude  of  the  point  r,  we  consider  Cr  to  be  a 
sine  of  an  arc,  and  C  P  the  radius. 

Therefore,    3261"  :  1324".3  :  :  R  :  sin.  x  =  23  58  00 
To  this  add  the  sun's  declination,  21  11  43 

Sum  is  latitude  where  the  sun  will  be 

centrally  eclipsed  on  the  meridian,  -  45  9  43  N. 
HOW  to  find  Wherever  the  sun  is  centrally  eclipsed  on  the  meridian,  it 
the  longitude  is  apparent  noon  at  that  place,  but  at  Greenwich  the  apparent 
pla°e  time  is  8  h.  57  m.  37  a.,  p.  M.  ;  this  difference,  changed  into  lon- 
>i> central-  gituole,  gives  134°  25'  west,  within  a  degree  of  the  result  de- 
ly  eclipsed  on  termined  from  the  projection;  and  it  is  not  important  to  go 

the  meridian.  .      ,  .          ,,          II-T  . 

over  a  trigonometrical  computation  for  tho  longitudes,  since 


ECLIPSES.  301 

i 

we  arc  sure  of  knowing  how  to  do  it;  and  wo  are  also  sure  CHAP.  IV. 
that  the  results  will  not  differ  much  from  those  already  de- 
termined. 

In  short,  from  the  elements,  the  figure,  and  a  knowledge  snfficiem 
of  trigonometry,  we  can  determine  all  the  important  points  in  * in  th« 
each  of  the  three  lines  c  d,  kq,  and  a  b,  for  between  them  we 
have,  or  may  have,  a  complete  net-work  of  plane  triangles. 


CHAPTER    V. 

LOCAL     ECLIPSES,     ETC. 

WE  now  close  the  subject  of  eclipses  by  showing  how  to    CHAP.  v. 
project  and  accurately  compute  every  circumstance  in  rela- 
tion to  a  local  eclipse. 

For  an  example,  we  take  the  eclipse  of  May,  1854,  and  for 
the  locality,  we  take  Boston,  Mass.,  because  we  anticipated  a 
central  eclipse  at  that  pjace,  but  the  result  of  computations 
shows  that  it  will  not  be  quite  central  even  there.  We  use 
the  same  elements  as  for  the  general  eclipse. 

THE    CONSTRUCTION. 

Draw  a  line  CD,  and  divide  it  into  65  *qual  parts,  and  The  scat* 
consider  each  part  or  unit  as  corresponding  to  one  minute  of 
the  moon's  horizontal  parallax.  From  C,  as  a  center,  at  a 
distance  equal  to  the  diff.  of  parallax  of  the  sun  and  moon 
(54  21 ),  describe  a  semicircle  north  or  south  according  to 
the  latitude,  or  describe  a  whole  circle  if  the  latitude  is  near 
the  equator. 

From  C  draw  C<s>,  the  universal  meridian,  at  right  angles 
to  CD,  and  from  05  take  25  qp  and  05  =£=,  each  equal  to  the 
obliquity  of  the  ecliptic  (  23°  27' )  and  draw  the  straight  line 
«f^,  =^  on  the  right.  Subtract  the  sun's  longitude  from 
90°  or  270°  to  find  its  distance  from  the  nearest  solstitial 
point,  and  note  the  difference  ( in  this  example  24°  46' ).  th^Tx"/ 

From  the  point  a,  with  a  T  as  radius,  make  a  G  equal  to  ecliptic. 


302  ASTRONOMY 

CHA*»  v-  the  sine  of  24°  46',*  and  join  C  G,  and  produce  it  to  E\  CE 
is  the  axis  of  the  ecliptic :  this  line  is  variable,  and  is  on  the 
other  side  of  the  line  C<s>  between  June  20  and  Decem- 
ber 21. 

How  to  find  From  E  take  the  arc  EL  equal  to  the  moon's  visible  path 
the  moon'g  with  the  ecliptic,  to  the  right  of  E  when  the  moon  is  descend. 
orbit  ing,  but  to  the  left  when  ascending  as  in  the  present  exam- 

ple.    Join  C L,  a  line  representing  the  axis  of  the  moon's  orbit. 
To  and  from  the  reduced  latitude  of  the  place  add  and  sub- 
tract the  sun's  declination: 

Thus,  Boston,  reduced  latitude,  -        42°    6'  39"  N. 

Sun's  declination,        -        21    11  43    N. 
Sum  is  63°  19-  22",  and  difference  is  20°  54'  56". 

HOW  to  find  From  (7,  make  (712  equal  to  the  sine  of  the  difference  of 
uHriiH  I"  the  two  arcs  (20°  54'  56")>  and  Cd  the  sine  of  the  sum 

miking    the  (63°  19'  22"). 

If  the    fcce      Divide  (12)  <J  into  two  equal  parts  at  the  pointy;  and  on 
over"  Pthl^  (12),  as  radius,  mark  the  sine  of  15°,  30°,  45°,  60°,  75°, 
•arth'iduc.   9Qo.  the  iine  -^  5^  runs  through  the  first  point;  8,  4,  through 
the  second,  &c. 

Subtract  the  latitude  (42°  6'  39")  from  90°,  thus  finding 
the  co-latitude  (47°  53'  21").  On  the  semidiameter  of  the 
earth's  disc,  as  radius,  take  the  sine  of  the  co-latitude  (47° 
53'),  and  set  off  that  distance  from  g,  both  ways  to  6 ;  thus 
making  a  line,  6,  6,  at  right  angles  to  the  universal  meridian, 
Cg.  On  g  (6)  as  radius,  and  from  the  pointy  as  a  center, 
find  the  sine  of  15°,  30°,  45°,  &c.,  and  set  off  those  distances 
each  way  from  g  and  through  the  points  thus  found,  draw 
lines  parallel  to  g  C;  these  lines,  meeting  the  lines  drawn  par- 
allel to  6^6,  will  define  the  points  5,  6,  7,  8,  &c.  to  12,  and 
1,  2,  3,  &c.  to  7,  the  hours  of  the  day  on  the  elliptic  curve. 
That  is,  our  supposed  observer  at  the  moon  would  see  Boston 
of  the  hours  (or  anj  other  place  in  the  same  latitude  as  Boston),  at  the 
ionnd  the  el-  point  9  when  it  is  9  o'clock  at  the  place,  and  at  12  when  it 
is  noon  at  the  place,  &c. 

*  The  reader  is  supposed  to  understand  how  to  draw  a  sine  to  any 
arc,  corresponding  to  any  radius,  either  with  or  without  a  sector 


ECLIPSES 


303 


304  ASTRONOMY. 

CHAP.  v.  As  this  curve  touches  the  disc  before  5  and  after  7,  it 
shows  that,  in  that  latitude,  on  the  day  in  question,  the  sun 
will  rise  before  5  in  the  morning,  and  set  after  7  in  the  eyen 
ing.  If  the  declination  of  the  sun  had  been  as  much  south  as 
now  north,  the  point  d  would  have  been  12  at  noon,  and  all 
the  hours  would  have  been  on  the  upper  part  of  the  ellipse, 
which  is  not  now  represented. 

From  C,  as  in  the  general  eclipse,  set  off  the  distance  C  n 
equal  to  the  moon's  latitude,  and,  through  the  point  n,  draw 
the  moon's  path  at  right  angles  to  CL. 

As  the  ellipse  represents  the  sun's  path  on  the  disc,  and  as 
the  point  (12)  refers,  of  course,  to  apparent  noon,  and  not  to 
mean  noon,  therefore,  we  will  mark  off  the  time  on  the  moon's 
path  corresponding  to  apparent  time. 

HOW  to  mark      When  the  moon's  center  passes  the  point  n,  it  is  at  ecliptic 
moon's11  JtT  conJuncti°n>  apparent  time,  at  Boston,  or  it  must  be  considered 
the  apparent  time  corresponding  to  any  other  meridian  for 
which  the  projection  may  be  intended. 

The  ecliptic  £  ,  apparent  time,  Greenwich,  is  8h.  49m.  Os. 
For  the  longitude  of  Boston,  subtract  4  44  16 

Conjunction,  apparent  time,  at  Boston,          4       4       44 

The  moon's  hourly  motion  from  the  sun  is  27   39":  take 

this   distance  from  the  scale,  in  the  dividers,  and  make  the 

small  scale  ab,  which  divide  into  60  equal  parts;  then  each 

in  this  case,  Pai>*  corresponds  with  a  minute  of  the  moon's  motion  from  the 

the     ellipse  SUI1)  an(j  fcDe  distance  ab  will  correspond  with  one  hour  of  the 

men!*  °°C  moon'8  motion  along  its  path.     At  4  h.  4  m.  44  s.  the  moon's 

tween  4  and  center  will  be  at  the  point  n;  the  sun's  center,  at  the  same 

$  o'clock.      time^  wyj  be  jugt  jjey0n^  the  p0int  4  on  the  ellipse  ;  and,  as 

the  distance  between  these  two  points  is  greater  than  the  sum 
of  the  semidiameters  of  sun  and  moon,  therefore  the  eclipse 
will  not  then  have  commenced ;  but  the  moon  moves  rapidly 
along  its  path,  and,  at  5  o'clock,  the  center  of  the  moon  will 
be  at  the  point  marked  5  on  the  moon's  path,  and  the  cente* 
of  the  sun  will  be  at  the  point  marked  5  on  the  ellipse;  and 
these  two  points  are  manifestly  so  near  each  other,  that  the 
limb  of  the  moon  must  cover  a  part  of  that  of  the  sun,  show- 


ECLIPSES.  30& 

ing  that  the  eclipse  must  have  commenced  prior  to  that  time.    CHAP.  v. 

To  find  the  time  of  commencement  more  exactly,  let  the  hour    TO  find  *• 

on  the  moon's  path  be  subdivided  into  10  or  5-minute  spaces,  aaore   MM| 

and  take  the  sum  of  the  semidiameter  of  the  sun  and  moon 

in  your  dividers  from  the  scale  CD,  and,  with  the  dividers 

thus  open,  apply  one  foot  on  the  moon's  path  and  the  other 

on  the  sun's  path,  and  so  adjust  them  that  each  foot  will  stand 

at  the  same  hour  and  minute  on  each  path  as  near  as  the  eye 

can  decide.     The  result  in  this  case  is  4  h.  28  m.     The  end  of 

the  eclipse  is  decided  by  the  dividers  in  the  same  manner,  and, 

as  near  as  we  can  determine,  must  take  place  at  6h.  44m. 

To  find  the  time  of  greatest  obscuration,  we  must  look    How  tofin* 

,         ,         ,  ,.  MI*        Ul»  •«*«•  *T 

along  the  moon  s  path,  and  discover,  as  near  as  possible,  from  grealegt  «^. 

what  point  a  line  drawn  at  right  angles  from  that  path  will  »curatio« 
strike  the  sun's   path    at  the  same  hour    and   minute;  the 
time,  thus  marked  on  both  paths,  will  be  the  time  of  great- 
est obscuration. 

In  this  case  it  appears  to  be  5  h.  40  m.,  and  the  two  cen- 
ters are  very  nearly  together ;  so  near,  that  we  cannot  decide 

on  which  side  of  the  sun's  center  the  moon's  center  will  be, 

. 
without  a  trigonometrical  calculation. 

To  show  a  representation  of  an  eclipse  at  any  time  during  ROW  to  fiM 
its  continuance,  we  must  take  the  semidiameter  of  the  sun  in  lhe    ***&&' 

tude   of   the 

the  dividers  from  the  scale ;  and,  from  the  point  of  time  on  eciip»e. 
the  sun's  path,  describe  the  sun ;  and,  from  the  same  point  of 
time  on  the  moon's  path,  describe  a  circle  with  the  radius  of 
the  moon's  semidiameter ;  the  portion  of  the  sun's  diameter 
eclipsed,  measured  by  the  dividers,  and  compared  with  the 
whole  diameter,  will  give  the  magnitude  of  the  eclipse  as  near 
as  it  can  be  determined  by  projection. 

The  result*  of  this  projection  are  as  follows: 

A  pp.  time.  Mean  time. 

Beginning  of  the  eclipse,  P.  M.,     4h.  28m.       4h.  24m.  39s. 
Greatest  obscuration,  5      40          5      36       39 

End  of  the  eclipse,  6      44          6      40      39 

' 

From  the  projection  the  two  centers  are  nearer  together 
than  the  difference  of  the  semidiameter  of  the  sun  and  moon 
20 


ASTRONOMY. 

v.  and  the  moon's  diameter  being  least,  the  eclipse  will  be  an- 
nular, as  represented  in  the  projection. 

The  above  results  are,  probably,  to  be  relied  upon  to  within 
three  minutes. 

We  have  now  done  with  the  projection,  as  far  as  the  particu- 
lar locality,  Boston,  is  concerned ;  but,  in  consequence  of  the 
facility  of  solution,  we  cannot  forbear  to  solve  the  following 
problem :  In  the  same  parallel  of  latitude  as  Boston,  find  the 
longitude  where  the  greatest  obscuration  will  be  exactly  at  2  p.  M. 
apparent  time. 
A  very  easy  From  the  point  2,  in  the  ellipse,  draw  a  line  at  right  an- 

•  ant  obtem  ^es  *°  ^e  moon's  Patn»  an(^  tna*  point  must  also  be  2h.  on 
the  moon's  path;  running  back  to  conjunction,  we  find  it 
How  solved,  must  take  place  at  1  h.  10  m. ;  but  the  conjunction  for  Green- 
wich time  is  8  h.  49  m.,  the  difference  is  7  h.  39  m.,  correspond- 
ing to  114°  45'  west  longitude ;  we  further  perceive  that  the 
sun  would  there  be  about  9  digits  eclipsed  on  the  sun's  north- 
ern limb. 

HOW  to  find      Now,  admitting  this  construction  to  be  on  mathematical 
•ore    accu-  princjpieg  ( ag  it  really  is,  except  the  variability  of  the  ele- 
ments ),  we  can  determine  the  beginning  and  end  of  a  local 
.eclipse  to  great  accuracy,  by  the  application  of  ANALYTICAL 

GEOMETRY. 

enerai      -^  fc  £,  p      ^  (7  05  be  two  rectangular  co-ordinates,  then 

equations  to 

aid  in  com-  the  distance  of  any  point  m  the  projection  from  the  center 
putingaiithe  can  ^e  determined  by  means  of  equations. 
oes  of  an  Let  x  and  y  be  the  co-ordinates  of  any  point  on  the  sun's 
eclipse  as  :path  or  elliptic  curve,  and  Jfand  Y  the  co-ordinates  of  any 
•n«npiac«any  P°*nt  on  *^e  m°on's  path,  then  we  have  the  following  equa- 
tions : 

( 1 )     y=p  sin.  L  cos.  D+p  cos.  L  sin.  D  cos.  t  (      solar 
(  2  )     x=p  cos.  L  sin.  t  I  co-ordin. 

(3)  Y=d±his\n.B       |  lunar  co.ordinates. 

(4)  X=hieos.B  ( 

In  these  remarkable  equations,  p  is  the  semidiameter  of  pro- 
jection, L  the  latitude,  D  the  sun's  declination,  t  the  time 
from  apparent  noon,  d  the  difference  in  decimation  between 


ECLIPSES.  $07 

iun  and  moon  at  the  instant  of  conjunction  in  right  ascen-   Cm*.  T. 
sion,  h  the  moon's  hourly  motion  from  the  sun,  i  the  interval 
of  time  from  conjunction  in  right  ascension  —  minus,  if  before 
conjunction  —  plus,  if  after;  and  B  is  the  angle  L  C  s,  or  the 
angle  which  the  moon's  path  makes  with  C  D. 

In  the  equations,  x  and  X  arc  horizontal  distances.  In 
equation  (  1  ),  the  plus  sign  is  taken  when  the  hours  are  on 
the  upper  side  of  the  ellipse,  as  in  winter  ;  when  on  the  lower 
side,  take  the  minus  sign. 

In  equation  (  3  ),  the  plus  sign  is  taken  when  the  motion  of  ExPu»atio» 
the  moon  is  northward,  and  the  minus  sign  when  southward.  of  the  *ym- 
The  sin.  t,  or  cos.  t,  means  the  sin.  or  cos.  of  an  arc,  corre- 
sponding to  the  time  at  the  rate  of  15°  to  one  hour. 

The  solar  and  lunar  co-ordinates,  or  equations  (  1  ),  (  2  ),   The  symbol 

(  3  ),  and  (  4  ),  are  connected  together  by  the  following  equa-         <4     , 

tions  ;  the  minus  sign  applies  to  forenoon,  the  plus  sign  to  time  of  eon- 

afternoon  :  junction    i. 

<4—  e=—  /;  ri*ht  M0— 

•ton. 


To  apply  these  equations,  and,  of  course,  the  former  ones, 
e,  the  interval  of  time  from  conjunction  must  be  assumed,  and, 
as  the  time  of  conjunction  is  known,  t  thus  becomes  known; 
(/,  h,  and  B,  are  known  by  the  elements;  therefore,  xt  y,  and 
X,  Y,  are  all  known.  But  the  distance  between  any  two 
referred  to  co-ordinates,  is  always  expressed  by 


When  an  eclipse  first  commences,  or  just  as  it  ends,  this  ex- 
pression must  be  just  equal  to  the  semidiameter  of  the  sun 
and  moon ;  and  if,  on  computing  the  value  of  this  expression, 
it  is  found  to  be  less  than  that  quantity,  the  sun  is  eclipsed ; 
if  greater,  the  sun  is  not  eclipsed ;  and  the  result  will  show 
how  much  of  the  moon's  limb  is  over  the  sun,  or  how  far 
asunder  the  limbs  are,  and  will,  of  course,  indicate  what 
change  in  the  time  mu§t  be  made  to  corresp  ind  with  a  con- 
tact, or  a  particular  phase  of  the  eclipse. 

For  an  eclipse  absolutely  central,  and  at  the  time  of  being 
central,  the  last  expression  must  equal  zero;  and,  in  that 


308 


ASTRONOMY. 


CBAP.  V. 


Application 

•f  the  preced- 
ing    eiprei* 


computation 
for  the  begin- 
ning  of  the 
eclipse  as 
•een  from 
Beiton. 


case,  x=X,  and  y=  Y.  In  cases  of  annular  eclipses,  to  find 
the  time  of  formation  or  rupture  of  the  ring,  the  expression 
must  be  put  equal  to  the  difference  of  the  semidiameters  of 
sun  and  moon.  In  short,  these  expressions  accurately,  ef- 
ficiently, and  briefly  cover  the  whole  subject;  and  we  now 
close  by  showing  their  application  to  the  case  before  us 

By  the  projection  we  decided  that  the  beginning  of  the 
eclipse  would  be  at  4h.  28m.,  apparent  time  at  Boston.  Call 
this  the  assumed  or  approximate  time,  and  for  this  instant  we 
will  compute  the  exact  distance  between  the  center  of  the  sun 
and  the  center  of  the  moon,  and  if  that  distance  is  equal  to 
the  sum  of  their  semidiameter,  then  4  h.  28  m.  is,  in  fact,  the 
time,  otherwise  it  is  not,  &c. 

h.    m.    •. 

Conjunc.  in  R.  A.,  app.  time,  Boston,    4  13  21 
Assume  i  equal  to  15 

Therefore,  t  is  equal  to  4  28  21=67°  5'  15". 

jo=54'  21"=3261.  Reduced  lat.,  Z=42°  6'  38". 

/>=21°  11'  43";      d=Cr=1324".3; 

JB=16°0'58". 

p  3261    -    log.  3.513511 
L  42°  6' 38"   sin.  9.826437 
D  21  11  43    cos.  9.969583 
<  67  5  15 


log.  3.513511 
cos.  9.870315 
sin.  9.558149 
cos.  9.590288 


2039.1 
346.3, 

y=1692j8 


log.  3.309531   .  346.3   log.  2.532263 


p 
cos.  L 


3.513511 
9.870315 
sin^J  _  9.961303 
*=222&5  log.  3.348129 


For  Fand  X: 


Ai414".75 


114.5 
add  1324.3 

1438.8 

:264 


sin.  9.440775 

log.  2.617800 

2.058575 


-  cos.  9.982804 

-  log.^.617800 
398.6     2.600604 


ECLIPSES.  309 

Here  are  two  sides  of  a  right-angled  triangle,  and  the  hy-    CHAP.  V. 
pothenuse  of  that  triangle  is  1857*'. 8,  which  is  the  distance 
between  the  center  of  the  sun  and  moon  at  that  instant ;  but 
the  semidiaraeter  of  the  sun  and  moon  is  only  1853";  there-    Theeoiip* 
fore  the  eclipse  has  not  yet  commenced,  and  will  not  until  the  m' 
moon  moves  over  4".8;  which  will  require  about    9s.,  as  we 
determined  by  proportion,  because  the  apparent  motion  of  the 
moon  will  be  almost  directly  toward  the  sun. 

When  the  apparent  motion  of  the  moon  is  not  so  nearly  in 
a  line  with  the  sun,  as  it  is  in  this  case,  we  cannot  proportion 
directly  to  the  result  of  the  correction.  In  fact,  the  apparent 
motion  of  the  moon  is  on  one  side  of  a  plane  right-angled  tri- 
angle, and  the  distance  between  the  center  of  sun  and  moon 
is  the  hypothenuse  to  that  triangle,  and  the  variation  of  the 
moon  on  its  base  varies  the  hypothenuse,  and  the  computa- 
tion must  be  made  accordingly. 

Hence,  to  the  assumed  time  of  beginning,    4  h.  28  m.  21  s. 
Add  ...    9 

Beginning,  apparent  time,          -         -         4      28       30 
Mean  time,  -         -        -         -         4      25       19 

By  the  application  of  the  same  expressions,  we  learn  that 
the  greatest  obscuration  will  take  place  at  4  h.  41  m.  mean 
time  at  Boston ;  and  the  apparent  distance  of  the  moon's  cen-  °f  the  ••"»' 
ter  will  be  18"  north  of  the  sun's  center ;  and,  as  the  moon's  "onj*n.t*™ 
semidiameter  is  57"  less  than  that  of  the  sun,  a  ring  will  be 
formed  of  between  10"  and  11"  wide  at  the  narrowest  point. 
End  of  the  eclipse,  6  h.  46  m.  58  s.  mean  time. 

In  computing  for  the  end  of  the  eclipse,  we  assumed 
t=2h.  33m.,  and  as  t  is  more  than  6h.,  the  second  part  of 
y  changes  sign,  as  we  see  by  the  figure ;  the  sun  after  6,  must 
be  above  the  line  6^6. 

Occultations  of  stars  are  computed  on  the  same  principle* 
as  an  eclipse  of  the  sun,  the  star  having  neither  diameter  nor 
parallax. 

From  forty  to  fifty  occultations  of  the  fixed  stars  by  the 
moon,  occur  each  month,  but  not  more  than  three  or  four 
ire  visible,  as  seen  from  any  one  place.  Very  few,  if  any, 
cave  Aldebaran,  are  visible  to  the  naked  eye. 


310  APPENDIX. 


APPENDIX  TO  ECLIPSES,  AND  OTHER  MATTER. 


introductoiy  WE  have  thus  far  treated  solar  eclipses  in  a  genera] 
Remarki.  'manner,  for  the  benefit  of  those  who  are  not  well  versed 
in  the  principles  of  spherical  sections,  but  now  we  propose 
to  show  the  strict  geometrical  principles,  which  cover  the 
whole  subject,  and  define  the  latitudes  and  longitudes 
which  bound  the -visibility  of  solar  eclipses  on  the  earth. 

We  shall  take  the.  same  eclipse,  as  before,  for  an  example, 
and  take  the  elements  as  we  find  them  in  the  English  Nau- 
.    tical  Almanac,  for  1854,  which  are  as  follows  : 

1854,  MAY  26.  h.   m.    s. 

Greenwich,  mean  time  of  tf  R.  A.         -          8  55  43.2 
Q  and  Q)'s  Right  Ascension,     -         -         413     7.41 

Q)'s  Declination  N.                   -  21°  33'  31"8 

9's  Declination  N.    -  21°  11'  16"8 

Q)'s  Horary  motion  in  R.  A.  31'  18"9 

^'s  Horary  motion  in  R.  A.  2'  31  "8 

Q)'s  Horary  motion  in  Decl.  N.  8'    7"3 

Q's  Horary  motion  in  Decl.  N.  25"9 

Q)'s  Equatorial  Hor.  Parallax,  54'  32"6 

Equatorial  Hor.  Parallax,  8"5 

Q)'s  true  semidiameter,  14'  53"6 

Q's  true  semidiameter,              -  15' 48"9 

From  these  elements,  the  following  results  have  beeii 
comDuted,  which  we  extract  from  the  Nautical  Almanac. 


APPENDIX. 


31 


LINE  OF  CENTRAL  AND  ANNULAR  ECLIPSE. 


fORTl 

Longi  ude.        Latitude. 

Longitude. 

Latitude. 

*TAC>1  ^nr 

51°  53'W      36J  18'N 
73°  53'          44°  14' 
92"  40'          48°  3J' 
11  6°  24'          49°  23' 
134°  45'  W      45°33'N 

143J  41'W 
156°  56' 
169°  28'W 
179°  24'  E 
162°  51'  E 

SOUTHERN  LIN 

41°  37'  N 
32°  39' 
22°  33' 
14°  52' 
6°43'N 

E  OF  SIMPLE  CO. 

IEBN  LlNE  OF  SIMPLE  CONTACT. 

Longitude. 

Latitude. 

I     Longitude. 

Latitude. 

••<.««» 

40°  16'  E 
48°     4' 
72°    4' 
90°  54' 
120°  28' 
130°  17' 
125°  43' 
116°45'E 

68°  57'  N 
76°  27' 
79°  57' 
80°    4' 
76°    0' 
65°  37' 
57°  42' 
52°  20'  N 

68°     2'W 
87°  44' 
101°  53' 
108°  24' 
126°  24' 
145°  54' 
163°  10'  W 
1  77°  22'  E 

5°  40'  N 
12°  49' 
16°  12' 
16°  49' 
13°  21'  N 
1°    7'  S 
14°  15' 
24°  16'  S 

ECLIPSE  BEGINS  AT  SUN-SET.                ECLIPSE  ENDS  AT  SUN-RISE.    • 

Longitude. 

Latitude. 

Longitude. 

Latitude 

>li-o/s«  «rfj  ni«J 
edi   1o   K*  J 

•      ••              - 
• 

38°  21'  E 
10°  15' 
2°  44'  E 
6C  54'  W 
40°  48' 
54°  23' 
65°  I8'W 

68°  55'  X 
65°  22 
63°  17' 
59°  29' 
29°  37' 
13°  29' 
6°    9'N 

174°36'E 
157°  21' 
150°  39' 
143°  48' 
124°  19' 
116°  50' 
117°    7'E 

23°  48'  S 
9°  43'  S 
0°  14'S 
10°  56'  N 
41°  16'N 
50°  37' 
51°  56' 

We  now  propose  to  show  most  clearly  to  the  geometrical 
student,  how  such  like  results  can  be  obtained.  We  shall 
make  the  same  general  construction,  as  before,  but  enlarge 
it,  to  show  the  sections  of  the  sphere,  and  the  application 
of  spherical  trigonometry. 

To  make  the  following  projection  appear  natural,  the 
learner  must  conceive  his  eye  to  be  in  a  line  between  the 
center  of  the  earth  and  the  center  of  the  sun,  and  at  a  dis- 
tance from  the  earth  equal  to  that  of  the  moon. 

The  diameter  of  the  earth  will  then  appear  to  cover  & 
space  in  the  heavens  equal  to  the  moon's  horizontal  par- 
allax, (£4'  52".6,  )  and  as  the  sun's  declination  is  north,  in 


B*«eofu* 


APPENDIX. 

this  case,  the  north  pole  of  the  earth  will  be  visible,  as 
represented  in  the  projection  at  the  point  P. 

In  eclipses,  we  suppose  the  sun  to  be  stationary,  and  the 
moon  to  move  with  the  excess  of  motion  between  the  SUQ 
and  moon,  therefore  we  subtract  the  parallax  of  the  sun, 
8". 5  from  (54'  32".6),  and  we  have  54'  24".  1,  or  3264".!, 
for  the  value  of  CB  or  CA. 

As  the  earth  is  not  a  perfect  sphere,  those  who  desire  to 
thewrfaceofbe  extremely  accurate,  would  subtract   11"  from  3255"!, 
th«  earth  as  making  3244".  1  for  the  value  of  CH.     Cd  would  be  about 
*  3259",  but  we  shall  attempt  no  such  accuracy.     Any  at- 
tempt to  correct  the  projection  from  a  true  circle,  would 
make  it  more  inaccurate  than  it  now  is.* 

The  perpendicular  plane  passing  through  CH  is  the  plane 
of  the  meridian,  that  meridian  on  which  the  sun  and  moon 
are  at  the  instant  of  conjunction  in  right  ascension.  Ob- 
serve that  CG  is  a  line  perpendicular  to  the  plane  of  the 
moon's  orbit. 

The  line  K,  m,  r,  d,  h,  is  the  plane  of  the  moon's  orbit, 
and  the  value  of  Cr  is  the  difference  in  declination  between 
the  sun  and  moon. 

The  inclination  of  the  moon's  path,  Kr  k,  to  the  meridian 
Howtoob-  ^^»  *8  determined  by  the  elements,  and  very  much  will 
tain  the  mo.  depend  on  the  accuracy  of  that  angle. 

tion    of   Uw       -ITII 


tion  to  tb«      Q)>s  motion  in  R.  A.  (per  hour,)     -     31'  18"9 
"""*  @'s  motion  in  R.  A.       -  -       2' 3I"8 

Q)'s  motion  from  the  sun,     -         -       28' 47"!  =  1727* 

But  the  moon  is  now  above  the  21st  degree  of  north  de- 
clination, where  the  length  of  a  degree  is  less  than  it  is  at 
the  equator,  in  the  proportion  of  cosine  of  the  declination 
to  the  radius. 

*  Here  we  would  caution  the  learner  not  to  assume,  at  the  outset, 
that  he  cannot  comprehend  the  figure,  because  it  appears  complex. 
It  is  a  diagram  which  includes  a  great  number  of  problems,  and  it  was 
drawn  with  a  design  to  solve  them  all : —  hence,  the  necessity  of  many 
lines  and  arcs. 


APPENDIX.  313 

Hence,  to  reduce  1727"!  to  its  equatorial  value,  or  to  Small  cir- 
reduce  it  to  a  great  circle,  we  must  multiply  by  cosine  of  ^"J"1  ^ 
the  moon's  declination,  and  divide  by  the  radius.  large  ones 

Thus         log.    1727"!  3.237292 

cos.   21°  33' 32"       -         -     9.968503 


1606"2  3.205795 

By  the  given  elements,  we  have,  for  hourly  increase  of 
the  moon's  declination,         -         -         -     8'    7"3 
And  of  the  sun's  -      25"9 

The  excess  is  -  -       7'    41"4=461"4 

Conceive  a  plane  triangle,  whose  base  is  1606.2,  and  Howtoind 
perpendicular  461.4,  and  find  the  acute  angles,  and  the  [^  ™hJJJj 
greater  of  which  (73°  58'  20")  is  the  angle  at  r,  in  the  axis  of  the 
figure,  and  the  less  (16°  1'  40")  is  the  angle  m  Cr,  which  |™a™r" l* 
measures  the  arc  6fff.  the  earth. 

The  hypotenuse  of  this  triangle  (1671")  is  the  apparent 
motion  of  the  center  of  the  moon's  umbra  over  the  earth's 
disc,  per  hour. 

From  Q)'s  declination  N".       -         21°  33'  31"8 

Take  Q's  declination         -         -     21°  11'  16"8 


And  we  have   Cr         -         -         -  22'  15"  =  1335" 

We  must  now  compute  the  values  of  (7*71=1282",  mr=  $everai  ime. 
J68"5.     We  have  before  remarked,  that  Kcmd  represents  comPuted* 
the  line  of  central  eclipse  over  the  earth,  and  it  is  obvious 
that  the  extent  of  the  eclipse,  north  and  south  of  this  line, 
must  depend  on  the  apparent  semidiameters  of  the  sun  and 
moon. 

Their  sum  is  (14'  53"5)  added  to  (15   48"9)  =  1842"4. 

Hence,  we  take  mO  and  mt,  each  equal  to  1842"4,  and 
through  the  points  0  and  t,  draw  lines  a  b  and  ef,  each 
parallel  to  the  central  line. 

The  line  a  b  is  the  diameter  of  a  circular  section  of  the  pogitjOBOf 
sphere,  which  defines  the  northern  line  of  simple  contact. the  P°'C  «f 
The  line  ef  is  the  southern  line  of  simple  contact.  The 
three  lines  a  b,  cd,  and  ef,  are  diameters  of  small  circles 


APPENDIX, 


• 


APPENDIX.  315 

of  the  sphere  —  one  of  these  small  circles  is  represented  in 
the  figure  by  the  dotted  line  c  R  Q  d. 

The  point  C  is  at  the  center  of  the  plane  of  projection      The  eart>l 

*.    J  ,      th«    bate  of 

or  we  can  conceive  it  to   be  on  the  surface  or  the  earth,  pr0jeotion 
directly  under  the  sun  at  noon.     Then  we  should  conceive 
the  line  C  If  to  be  the  meridian  arc  of  a  great  circle,  C  H 
being  90°,  whence  the  arc  HP  is  equal  to  the  sun's  decli- 
nation. 

Whence  G  HP  is  a  right  angled  triangle,  on  the  surface 
of  a  sphere  ;  G  R  P  is  another  spherical  triangle,  and  PR 
is  the  co-latitude  of  the  point  R,  on  the  surface  of  the  earth. 

We  will  now  continue  the  computation  of  all  the  lines 
and  arcs,  that  we  shall  have  occasion  to  use. 

We  have  already         (7*71=1282" 
Add  m<9=1842"4 

Sum  (70=3124"4     Diff.    Ct=560"4. 

(7<?=:3264"4 


Diff.  is  G0=   140" 

Having  the  diameter  of  the  circle,  and  G  0,  we  find 

aO=  06=945"8,  which  is  the  sine  of  the  arc  a  G=  16°  50'. 

.  Having  Cm  (1282"),  and  Cd  (3264"4),  the  right  angled 

triangle  m  Cd  will  give  us  mrf=mc=3001"2,  and  the  angle 

mCd,  or  the  arc  Gd=Gc=66*  53'. 

Again,  CA=(7^=326-4"4+1842"4=5106"8,  and  from 
the  triangle  mCAwe  find  wiA=4942"3, — its  double  is 
.ATA=9884"6.  The  angle  m  C  A=75°  28',  or  the  arc  Gq= 
75°  28'=  £.4. 

Observe  that  Kc=dh=mh—  mrf=4942"3  — 3001  "3= 
1941".  Also  observe  that  rtf  =  3001"3— 368"5=2632"8, 
and  cr=3369"8. 

Now  observe  the  arcs :  Computation 

The  arc  #£=90°.     From  the  values  of  Ct  (560"4)  and"fnrc  onth8 

base    of  MO* 

CTZ=(3264"4),  we  found  the  arc  Le  to  be  9°  53'. 
But  OH    -  =:6°    lf  40" 

GL        -        -     =90° 


Whence  the  arc         He     -         -       =115°  54'  40" 


31  (J  APPENDIX. 

The  angle  G  C  JST=75°  28' ;  to  this  add  ##",  16°  1'  40', 
and  we  have  H  C ^=91°  29'  40",  or  D  C  K=\°  29'  40". 

We  have  now  the  arcs  ##,  //a,  #c,  77^1,  #e.  We 
have  also  the  arc  HP,  which  is  equal  to  the  sun's  decima- 
tion, and  the  angle  at  H &  right  angle. 

We  will  now  solve  the  spherical  triangle  GHP,  as  fol- 
lows: 

As     R  :  cos.  GH  :  -.  COB.  HP  :  cos.  G P. 
cos.  HP         21°  11'  16"  9.969604 

cos.  GH         16°     1'  40"  9.982782 


Lunar  pole  from  P,  26°  21'  30"     cos.  9.952386 

The  angle  H  G  P=56°  26'  27",  and  the  angle  G  P  H= 
38°  26'  35".  Consequently  the  angle  P  G  (7=33°  33'  33", 
and  the  angle  GPC=  141°  33'  25". 

The  foregoing  is  only  a  proper  preparation  for  solving 
the  many  propositions,  that  may  be  proposed  in  relation  to 
this  eclipse. 

G«n«r»i         l.  The  point  e,  is  the  most  southern  point  on  the  earth, 
<*"*itioili'   where  the  southern  line  of  simple  contact  first  touches  the 
earth.      What  is  the  latitude  and  longitude  of  that  point? 

2.  The  point  A,  is  the  point  where    the   eclipse  first 
touches  the  earth .      What  is  the  latitude  and  longitude  of 
that  point? 

3.  The  point  c,  is  the  point  where  the  central  eclipse 
first  comes  on    the   earth.      What  is  the  latitude  and  longi- 
tude of  that  point? 

4.  The  point  (a),  is  the  point  where  the  northern  line  of 
simple  contact  first  touches  the  earth.      What  is  the  latitude 
and  longitude  of  that  point? 

5.  The  point  (b),  is  the  point  where  the  northern  line 
of  simple  contact  passes  off  of  the  earth  into  open  space. 
What  is  the  latitude  and  longitude  of  that  point? 

6.  The  point  (d),  is  the  point  from  whence  the  central 
eclipse  leaves  the  earth,  and  rises  up  into  space.      What  it 
the  latitude  and  longitude  of  that  point? 


APPENDIX.  317 

7.  The  point  (q),  is  the  point  on  the  earth  where  the 
eclipse  finally  leaves  the  earth.      What  is  the  latitude  and 
longitude  of  that  point? 

8.  The  point  /,  is  the  point  where  the  southern  line  of 
simple  contact   finally  leaves  the  earth.      What  is  the  lati- 
tude and  longitude  of  that  point? 

We  will  answer  each  of  these  eight  questions,  in  order. 
First  in  respect  to 

LATITUDE. 

To  each  question  of  latitude,  there  exists  a  correspond-      Latitndet 
ing  right   angled   spherical   triangle,   the    hypotenuse  of  "^J 
which  is  a  co-latitude.  eclipses. 

To  the  first  question  —  the  latitude  of  the  point  e,  —  we 
have  the  arc  H  G  e  for  one  side  of  the  triangle,  and  HP  for 
the  other  side.  HP  is  common  to  all  the  triangles. 

To     cos.ffP      21°  11'  16"  9.969604  HOW 

Add  cos.  He     115°  54' 40"     -         -    9.640457 


Sum  is  cos.  Pe,  or  sin.  24°  4'  9.610061 

Thus  we  learn  that  the  most  southern  point  of  the  eclipse 
was  in  24°  4'  south  latitude. 

To  -  9.969604 

Add  cos.  AH  91°  29'  40"   -     £***      8.416319 


cos.    91°  24'      -         -  8.385923 

This  result  shows  that  the  eclipse  first  touches  the  earth 
in  latitude  1°  24'  south. 

To  -    9.969604 

Add  cos.ffc  82°  54' 40"     -         9.091347 


cos.  83°  23',  or  sin.  6°  37'       9.060951 
The  central  eclipse  commences  in  latitude  6°  37'  north. 
To  -    9.969604 

Add  cos.  Ha  32°  51'  40"  9.924302 


cos.  38°  26'  26",  or  sin.  51°  33'  24"  9.893906 
A  result  showing  that  the  northern  line  of  simple  con 
tact  first  touches  the  earth  in  latitude  51°  33'  24"  north. 


318  APPENDIX. 

To  -  9.969604 

Add  cos.  Hb  48'  40"         -  9.919956 


cos.  21°  12',  or  sin.  68°  48'      9.969560 
And  this  last  result  shows  that  the  northern  line  of  sim- 
ple contact  leaves  the  earth  in  latitude  68°  48'  north. 

To  -         -    9.969604 

Add  cos.  Hd  50°  51'  20"     •        9.800221  (  Gd—GH) 

sin.  36°  3'  9.769825 

The  central  eclipse  leaves  the  earth  in  latitude  36°  3'  N. 

To  ...    9.969604 

Add  cos.  59°  26'  20"  9.706255  (Gq—Gh) 

sin.  28°  16'  9.675859 

This  result  indicates  that  the  eclipse  finally  leaves  the 

earth  in  latitude  28°  16'  north. 

The  arc    Ge=  #/=99°  53',   from  which  subtract  Off 

16°  1'  40",  and  we  have  83°  51'  20"  for  the  value  of  the 

arc  Sf. 

To  -    9.969604 

Add  cos.  83°  51'  20"          -          9.029832 


sin.  5°  44'  20"  8.999136 

Whence  we  learn  that  the  southern  line  of  simple  con- 
tact terminates,  on  the  earth,  in  latitude  5°  44'  north. 

Thus  we  have  determined  the  latitudes,  of  all  the  points 
of  the    beginnings  and  endings  of    the  eclipse,   on  these 

Howtoob.threelines- 
»intheiati.      The  latitudes  of  points  on  the  central  line  across  the 

t0tnt  °L  theeart^'  can  ke  determined  by  spherical  triangles, 
•mail  circlet  Thus,  the  latitude  of  the  point  R,  on  the  surface  of  the 
•fth«moon'«  earth,  is  determined  by  the  spherical  triangle  GRP, 
GR=66°  33',  #P=26°  21'  30",  and  the  angle  g  OR,  or 
R  G  P,  may  be  assumed.  Then  the  third  side,  RP,  is  the 
co-latitude  of  R,  and  R  may  represent  any  point  on  the 
small  circle  c  R  Q  d. 

Observe  that  P  Zt  is  the  co-latitude  of  the  point  Z,  and 


APPENDIX.  319 

PQ  is  the  co-latitude  of  the  point  Q,  and  that  point  is  the 
highest  latitude  on  the  line  of  the  central  eclipse. 

In  the  same  manner,  the  latitudes  of  points  on  the  lines 
ab  and  ef  may  be  determined,  and  the  latitudes  of  all  points 
on  other  lines  parallel  to  these,  if  we  know  the  number  of 
degrees  from  G,  as  we  know  the  distance  GR  (66°  53'). 

If  GR  terminated  in  the  line  ab,  its  value  would  be  16° 
50'.  If  continued  to  ef,  its  value  would  be  115°  54'  40". 

The  latitude  of  the  point  Q  can  be  determined  thus  : 
PQ=(GQ—GP).     Lat.  Q=90°—  (  GQ—  GP)=\}6°  21' 
30"—  66°  53'=49°  28'  30",  which  is  the  highest  latitude  of 
the  central  eclipse. 

The  latitude  of  the  point  Z,  (where  the  sun  is  centrally 
eclipsed  at  apparent  noon,)  is  determined  by  the  right 
angled  triangle  GHZ,  as  follows  : 

Rad.  :  cos.  Gff  :  :  cos.  HZ  :  cos.  GZ.  Hoxri»fi«d 

Whence  cos.  OZ= 


_ 

COS.  Gff  COS.  16°   1'  40"  ualeciipwmt 

lO.+cos.  66°  53'  -         19.593955 

cos.  16°    1'  40"       -         -      9.982781 


From       -         65°  53' 40"  9.611174 

Subt.  HP        21°  11' 16" 


PZ  or  co-latitude        44°  42'  24" 

Thus  we  determine  that  the  sun  must  be  centrally 
eclipsed,  at  noon,  in  latitude  45°  18'.  The  result  given  in 
the  Nautical  Almanac,  is  45°  33',  but  we  do  not  claim  the 
utmost  accuracy  in  computation,  our  object  is  to  teach 
principles.*  For  the  points  of  beginning,  we  agree  with 
the  Nautical  Almanac,  but  here,  and  at  the  end  we  differ  in 
latitude  a  few  miles.  We  have  considered  the  moon's  path 
as  a  straight  line,  but  it  is  slightly  curved,  —  it  does  not 

»  If  we  reduce  the  parallax  corresponding  to  45°«  6",  (see  table  on 
page  39  of  tables),  our  latitude  would  -hen  nearly  correspond  with  th« 
Nautical  Almanac. 


320  APPENDIX. 

change  its  declination  at  the  same  rate  at  every  hour  during 
the  eclipse,  as  this  projection  supposes,  and  this  will  ac- 
count for  part  of  the  difference  between  us  and  the  Nautical 
Almanac. 

LONGITUDES. 

Longitudes      The  longitude  where  the  sun  will  be  central!}'  eclipsed 

in   which  an     ,  ,  .        ,    ,          , 

•eiipie   mayat  aPParent  noon»  is  determined  by  the  apparent  time  of 
b«finor  end.  conjunction,  in  right  ascension,   at  Greenwich,  and  from 
this  longitude  we  obtain  all  others. 

h.  m.   s. 
The  mean  time  of  tf  at  Greenwich,  is  8  55  43.2 

Equation  of  time  at  this  moment,         -          -|-    3  15.4 
Apparent  time  of  tf  at  Greenwich,         -         8  58  58.6 

We  shall  call  this  8h.  59m.,  and  this  time  shows  that 
the  sun  will  be  eclipsed  at  apparent  noon,  on  that  meridian 
which  differs  from  the  meridian  at  Greenwich  by  8h.  59m., 
which  corresponds  to  134°  45'  west. 

Before  we  can  go  through  with  the  subject  of  longitudes 
we  must  be  able  to  determine  the   apparent  time  of  the 
rising  and  setting  of  the  sun,  in  any  latitude,  correspond- 
ing to  any  declination,  as  is  shown  in  the  following  table. 
The  construction  of  the  table  will  be  readily  understood 
twm  of  the  ^7  those  who   comprehend   spherical   trigonometry.     The 
t«M«.          triangle  referred  to  will  be  found  on  page  245.     It  is  CRS. 
The  angle  SCR  is  the  co-latitude,  and  is  given — and  the 
side  SR  is  the  declination,  and  is  given  — and  the  arc  CR, 
which  measures  the  angle  CPR  at  the  pole,  is  sought. 
This  arc,  changed  into  time,  and  six  hours  added,  produces 
the  results  found  in  the  table,  to  the  nearest  minute. 

Aa«am  !e      ^or  examP^6'  &**&  ^e  iime  °f  apparent  sun-set  corres- 
'ponding  to  sun's  declination  16°  N.  and  lat.  34°  N. 

tan.  16  9.457496         1 1°  9'=44m.  36s. 

tan.  34     -         -  9.828987         App.  time  6h.  44m.  36s. 


sin.  11°  9'     -       9.286483         In  table  6h.  45m. 


APPENDIX. 


32 


TABLE, 

Showing  tUe  apparent  time  the  sun  rises  when  the  latitude  and  declination 

are  on  opposite  sides  of  the  equator,  and  the  apparent  time  the 

sun  sets,  when  both  are  on  the  same  side  of  the  equator. 

DECLINATION. 


L*t. 

50  ;  90 

12° 

140 

16« 

17° 

180 

19^ 

20° 

21^ 

22° 

230 

240 

h.  m. 

h.  m. 

h.  m. 

h.  m 

h.m. 

h.m 

Ii7m~. 

h.  ra. 

h.  m 

h.m. 

h.w 

h.  m. 

h.  m 

h.  ra. 

40 

6  1 

6  1 

6  2 

6  3 

6  4 

6  4 

6  5 

6  5 

6  5 

6  6 

6  6 

6  6 

6  7 

6  7 

6 

6  1 

6  3 

6  3 

6  4 

6  6 

6  7 

3  7 

6  8 

6  8 

6  8 

6  y 

6  9 

6  10 

6  12 

8 

6  2 

6  4 

6  6 

6  7 

6  8 

6  9 

8  10 

6  10 

8  11 

6  11 

6  12 

6  13 

6  13 

6  14 

10 

6  2 

6  5 

6  6 

6  9 

6  10 

6  12 

6  13 

6  13 

6  14 

6  14 

6  15 

6  16 

6  17 

6  18 

12 

6  3 

6  6 

6  7 

6  10 

6  12 

6  14 

o  15 

6  16 

6  16 

6  17 

6  18 

6  19 

6  20 

6  22 

14 

6  3 

6  6 

6  9 

6  12 

8  14 

8  ll> 

3  17 

6  19 

6  20 

6  21 

6  22 

6  23 

6  24 

6  25 

16 

6  3 

6  7 

6  10 

6  14 

6  16 

6  19 

8  20 

6  21 

6  23 

6  24 

6  25 

6  27 

6  28 

6  29 

18 

6  4 

6  8 

6  12 

6  16 

6  19 

6  21 

o  23 

6  24 

6  26 

6  27 

6  29 

6  30 

6  32 

6  33 

20 

6  4 

6  9 

6  13 

6  18 

8  21 

6  24 

3  26 

6  27 

6  29 

6  30 

6  32 

6  34 

6  36 

6  37 

22 

6  5 

6  10 

6  15 

6  20 

6  23 

6  27 

8  28 

6  30 

6  32 

6  34 

6  36 

6  38 

6  39 

6  41 

24 

6  5 

6  11 

6  16 

6  22 

6  26 

6  29 

o  31 

6  33 

6  35 

6  37 

6  39 

6  41 

6  44 

6  4G 

26 

6  6 

6  12 

6  18 

6  24 

6  28 

8  32 

3  34 

6  36 

6  39 

6  41 

6  43 

6  45 

6  48 

6  50 

28 

6  6 

6  13 

6  19 

6  26 

6  30 

6  35 

3  37 

6  40 

6  42 

6  45 

6  47 

6  50 

6  62 

6  65 

30 

6  7 

6  14 

6  21 

6  28 

6  33 

6  38 

>  41 

6  43 

6  46 

6  49 

6  51 

6  54 

6  67 

7  0 

32 

6  8 

6  15 

6  23 

6  31 

6  38 

6  41 

3  44 

8  47 

6  50 

6  63 

6  56 

6  58 

7  1 

7  6 

34 

6  8 

6  16 

P  25 

6  33 

6  39 

3  45 

6  48 

6  51 

6  54 

6  67 

7  0 

7  3 

7  7 

7  10 

36 

6  9 

6  18 

6  26 

6  36 

6  4-2 

6  48 

3  51 

8  55 

6  58 

7  1 

7  6 

7  8 

7  12 

7  16 

38 

<>  9 

6  19 

6  28 

6  38 

6  45 

6  52 

3  55 

6  69 

7  2 

7  6 

7  10 

7  14 

7  17 

7  81 

40 

6  10 

6  20 

6  31 

6  41 

6  48 

6  56 

3  59 

7  3 

7  7 

7  11 

7  15 

7  19 

7  23 

7  28 

42 

6  1) 

6  22 

6  33 

6  44 

8  5-2 

7  0 

7  4 

7  8 

7  12 

7  17 

7  21 

7  25 

7  30 

7  36 

44 

6  12 

6  23 

6  35 

6  47 

d  66 

7  4 

7  9 

7  13 

7  18 

7  22 

7  27 

7  32 

7  37 

7  42 

46 

6  12 

6  25 

6  38 

6  51 

7  0 

7  9 

1  14 

7  19 

7  24 

7  29 

7  34 

7  39 

7  44 

7  60 

48 

6  13 

6  27 

6  41 

6  66 

7  4 

7  14 

7  19 

7  25 

7  30 

7  35 

7  41 

7  47 

7  63 

7  69 

50 

6  14 

6  29 

6  44 

6  59 

7  9 

7  20 

7  25 

7  31 

7  37 

7  43 

7  49 

7  55 

«'  2 

8  8 

62 

6  15 

6  31 

6  47 

7  3 

7  14 

7  26 

7  3-2 

7  38 

7  45 

7  61 

7  58 

8  5 

8  12 

8  19 

54 

6  17 

6  33 

o  50 

7  8 

7  20 

7  33 

7  40 

7  46 

7  63 

8  0 

8  8 

8  15 

8  23 

8  31 

56 

6  18 

6  36 

6  54 

7  13 

7  27 

7  41 

7  48 

7  65 

8  3 

8  11 

8  19 

8  27 

8  36 

8  46 

68 

6  19 

6  39 

6  59 

7  20 

7  34 

7  49 

7  57 

8  5 

8  14 

8  2:2 

8  32 

8  41 

8  51 

9  2 

60 

6  21 

6  42 

7  4 

7  26 

7  42 

7  59 

3  8 

8  17 

8  26 

8  36 

8  4< 

8  58 

9  9 

9  22 

61 

6  22 

6  44 

7  6 

7  30 

7  47 

8  5 

3  14 

8  24 

8  34 

8  44 

8  55 

9  7 

9  20 

9  34 

62 

6  23 

6  46 

7  9 

7  34 

7  62 

8  11 

S  20 

3  31 

8  42 

8  53 

9  5 

9  18 

9  32 

9  47 

63 

6  24 

6  48 

7  13 

7  39 

7  57 

8  17 

3  27 

8  38 

8  50 

9  2 

9  16 

9  30 

9  46 

10  4 

64 

6  25 

6  50 

7  16 

7  43 

8  3 

8  24 

3  35 

8  47 

9  0 

9  13 

9  28 

9  44 

10  2 

10  24 

65  6  26 

6  52 

7  19 

7  48 

8  9 

8  32 

8  44 

8  67 

9  10 

9  26 

9  42)10  1 

10  22 

.0  61 

The  sun  is  supposed  to  be  stationary  and  vertical  over 
the  point  C.  The  earth  revolves  from  west  to  east,  hence 
observers  on  the  curve  LA  eg  see  the  sun  in  the  eastern 
horizon  as  it  appears  to  them. 

When  the  moon  arrives  at  K,  an  observer  at  A  would 
see  its  limb  just  meet  the  limb  of  the  sun,  and  the  sun 
would  he  riLlng  to  that  observer. 

But  that  observer  is  in  latitude   1°  24'  south,  and  the 


322 


APPENDIX. 


declination  of  the  sun  is  21°  11'  north,  whence  by  the  table, 
the  sun  must  rise  about  two  minutes  after  six.* 
.    '        *  h.  m.  s. 

6     1  44 

3  10  36 


Sun  rises,  apparent  time, 
Time  from  .AT  to  r, 


Time  of  tf  at  .this  locality,  -  9  12  20  morning. 
Time  of  tf  at  Greenwich,     -  8  58  58  evening. 

Difference  of  meridians,         11  46  38=Lon.  176°  39' W. 

When  the  moon  arrives  at  c,  the  sun  is  centrally  eclipsed 
that  point,  and  it  is  the  first  point  on  the  earth  at  which 
*  L .'!  the  sun  is  centrally  eclipsed,  and  its  latitude  is  6°  37'  K". 
But  in  this  latitude,  when  the  sun  is  21°  11'  north, 

h.  m.  s. 
The  sun  rises  at  5  50  30 

Time  the  Q)  passes  from  c  to  r  is  2    0  57 


;  • 

Various 

ba  ftudg  ° 


App.  time  of  tf  at  this  locality,  7  51  27  morning. 
Time  of  cf  at  Greenwich,  8  58  58  evening. 


Difference  of  meridians 


13    7  31=Lon.  163°  7'  E. 


When  the  moon  arrives  at  d,  the  central  eclipse  passeg 
off  of  the  earth  ;  but  this  point  is  in  latitude  36°  3'  north, 
and  the  declination  is  21°  11'  north.  With  this  latitude 
and  declination,  we  enter  the  table,  and  find  that 

h.  m. 

The  sun  sets  at  -         -     7    5 

The  Q)  passed  from  r  to  d  m          1  35 


Time  of  of  at  this  place, 
Time  of       at  Greenwich, 


Difference  of  meridians, 


5  30  P.  M. 
8  59  P.  M. 

3  29=Lon.  52°15'W. 


*  When  we  wish  to  be  very  accurate,  we  must  not  use  the  table,  but 
aolve  the  problem  as  taught  on  page  246,  thus  : 

tan.    1P24'        -  8.388092 

tan.  21°  11'    -        -  9.588316 


sin.  26' 
This  arc,  26',  corresponds  to  1m.  44s. 


7.976408 


APPENDIX.  323 

When  the  moon  arrives  at  h,  the  eclipse  leaves  the  earth, 
and  the  last  observer  must  be  at  q  —  but  the  latitude  6f  q 
we  have  determine)  1  to  be  28°  16'  N.,  and  in  this  latitude 

h.  m.    s. 

The  sun  sets  at  -         6  49 

The  Q)  moves  from  r  to  h  in    2  44  42 

Time  of  tf  in  this  locality,       4    418  evening. 
Time  of  tf  at  Greenwich,         8  58  58  evening. 


Difference  of  meridians,  4  54  40=Lon.  73°  39'  W. 


In  the  same  way  we  can  determine  the  longitude  of  the    • 
extreme  points  on  the  northern  line  a  b,  and  on  the  south- 
ern line  ef. 

An  observer  at  (a)  will  not  see  the  moon  touch  the  sun 
until  the  moon  arrives  at  a  distance  from  m  equal  to  a  0. 
But  an  observer  at  a  sees  the  sun  in  his  horizon,  and  to 
him  it  is  rising.  Now  we  have  determined  the  latitude  of 
a  to  be  51°  33'  24",  and  in  this  latitude,  when  the  sun's 

declination  is  21°  11'  north, 

h.  m.  s. 
The  sun  rises  at  43       appa.  time  morn. 

The  Q)  moves  from  the  perpendic  - 

ular  let  fall  from  a  to  r*  ( 1314")  in     50  49 

The  time  of  tf  in  this  locality,  4  53  49  morning. 
The  time  of  tf  at  Greenwich,      8  58  58  evening. 

Difference  of  meridians,  (if  W.)    16    511 

(ifE.)       7  52  49=Lon.ll8°10'E 

The  point  5,  is  in  latitude  68°  48',  and  from  this  latitude, 
when  the  sun's  declination  is  21°  11'  north,  we  find  that 

h.  m.   s. 
The  sun  must  set  at  1 1  50  28  apparent  time. 

This  is  past  </  by  48"4,  ==  1  44 


.  »The  moon  moves  along  the  central  line  at  the  rate  of  1671"  per 
%our.  The  distance  aO,  is  945"8,  and  mr  is  368*5,  their  sum  is  1314"3, 
•which  divided  by  1671*.  produces  50m.  49s. 


APPENDIX. 

Time  of  tf  at  this  locality,     11  48  44  evening. 
Time  of  tf  at  Greenwich,        8  58  58  evening. 

Difference  of  meridians,  2  49  46=Lon.  41°  26  K. 

The  point  e,  is  in  latitude  24°  4'  south,  and  we  »%•?  by 
the  table,  that  the  sun  must  rise  in  that  latitude,  on  that 
day,  at  6h.  47m.  apparent  time. 

Bui  when  the  center  of  the  sun  is  projected  at  *,  the 
center  of  the  moon  must  be  near  c,  to  enable  the  Jimbs  to 
touch.  (The  distance  from  m  must  be  equa)  to  «/.) 

Not  to  be  very  precise,  we  may  say  that  the  time  re* 
quired  for  the  moon  to  pass  over  the  distance*  «t  and  mr, 

is  2h.  and  5m. 

h.  m. 

Hence,  to         -         -         -          6  47  A.  M 
Add  •         -         -      2    5 

Time  of  tf  for  this  locality,        8  52  A.  M 
Time  of  </  at  Greenwich,  8  59  P.  M. 

Difference  of  meridians  (W.),   12    7=Lor    178°  15' E. 

We  differ  lesd  than  half  a  degree  from  thfe  result  given 
in  the  Nautical  Almanac,  notwithstanding  Otir  rough  and 
summary  mode  of  computation. 

In  like  manner,  we  can  find  the  longitude  of /to  be  not 
far  from  68°  west. 

Lonsitu>     We  will  now  consider  any  other  point  on  the  circumfe- 
^^"'""^rence  of  projection  within  the  limits  of  the  eclipse,  and 
points  taken  determine  the  latitude  and  longitude  of  that  point,  which 
w^  a^so  ^e  a  P°int  where  the  eclipse  will  begin  at  sun-rise, 
if  the  point  is  on  the  western  side  of  the  projection,  or  a 
•f  projection,  point  where  the  eclipse  will  end  at  sun-set,  if  it  be  on  the 
eastern  side  of  the  projection. 

Let g be  a  point  taken  at  hap-hazard  on  the  circle  Ac  Ot 
and  suppose  the  arc  Qg  be  taken  equal  42°.  Then  OP 
will  be  the  co-latitude  of  the  point  g,  and  the  sun  being 
vertical  over  C,  is  90°  distant  from  the  observer/and  of 
course,  in  his  horizon. 


APPENDIX. 

We  take  the  distance  (gu),  (gr),  equal  to  mO,  the  sum 
of  the  semidiameters  of  the  sun  and  moon  (1842"4).  When 
the  moon  comes  to  F,  the  eclipse  commences,  and  when  it 
passes  u,  the  eclipse  ends. 

To  -      9.969604 

Add  cos.  59°  1'  40"  -  9.723738 

cos.  co-latitude  — or  sin.  lat.  29°  34'  30"  9.693342 

In  like  manner  we  may  find  the  latitude  of  any  point  on     A  *eneral 

.  investigation 

the  arc  within  the  limits  of  the  eclipse,  from  e  to  a,  and  Of  the  pro. 
from  b  to/.  Observe  that  the  points,(^')  and  (g),  are  at blem  for  find; 
equal  distances  from  F;  hence,  the  eclipse  will  be  seen  to  ["aVand  loo* 
commence  at  the  same  moment  of  absolute  time,  from  (^)gitud*  ofpu- 
and  (g').  The  difference  between  the  latitudes  of  these  °"jWteeiretbe* 
two  points  is  the  difference  between  the  two  arcs  Pg'  andgimandeiuu 
Pg,  and  their  difference  in  longitude  is  equal  to  the  arc  *^d  I*un^"* 


The  distances  ru  and  r  Fare  obtained  thus  : 
Observe  that  m  S  is  the  sine  of  Og  to  the  radius  cm. 
Let»iF=y.         Then  VS=y  —  cm  sin.  Og. 

Sff=cmco8.  Gg  —  Cm. 

(y—cm  gin.  Gg)2  -{-(cm  cos.  Gg—Cm)2=(  \  842"!)*. 
Whence 


y=  "^/(184r  fj'—  (cm  cos.  <fy—  <7m)2+cm  sin.  Gg.     (1) 
Let  mu=x.     Then 

(cm  sin.  Gg—  z)2  +(cm  COB.  Gg—  C'm)2=(1842"l)2 
Whence  __ 

x—cm  sin.  Gg—  ^(4842"!  )a~  (em  cos.  Gg—  Cm)a    (2) 
When  the  eclipse  begins  at  sun-rise,  the  time  of  con- 
junction at  that  locality  is  the  second  member  of  the  fol- 
lowing equation  ; 

(3) 


When  the  eclipse  ends  at  sun-rise,  the  time  of  ^  ia 

/  .      ,  a*-4-mr         /  .  \ 

/Y=sun-nse-|--n:  —         (4) 

r  v  j 


*  The  intelligent  pupil  must  be  aware  that  this  is  a  variable  ele- 
ment, different  in  different  eclipses,  and  it  varies  slightly  during  the 
tame  eclipse. 


APPE  NDIX. 

When  the  arc  is  on  the  other  side  of  GC,  we  have  the 
local  times  of  conjunction,  as  follows  : 

O^sun-set-Ai""^         (5) 

i.**..  W»V 

^nations,  and  /  /X-\-THT  \  /  rt » 

a      practical  CJ=SUn-S6t —i  j~r J  (6) 

application  of  \  *  O  /  1    X 

We  will  apply  this  to  the  assumed  angle  Q-g  42°,  and 
find  the  longitudes  in  question. 

cm  3001. 2  3.477260  cm  3.477260 

sin.  %420         9.825511  cos.  9.871073 


2008"  3302771          3230"  3.348233 

Cm  1282" 

948" 


)2—  (948")24-2008"=3587".3. 
ar=2008"—  1579"3=428"7. 

These  values  of  y  and  x,  placed  in  equations  (3)  and  (4), 
give  us,  at  the  place  of  beginning, 

h.  m. 

•Time  of  c/=5h.  10m.+?^3±:3^==:7  32  A.  M. 

1671" 

Time  of  tf  at  Greenwich,         -         -  8  59  P.  M. 

Difference  of  meridians,  (west,)     -         1327= 

Lon.l58°  15'B. 
Place  of  ending,  ^  m 


Time  of  c/=5h.  10m.+        -=5  38  A.  M. 

1671" 

Time  of  tf  at  Greenwich,  -        8  59  P.  M. 

Difference  of  meridians  (west),  15  21  = 

Lon.  129°4£/E. 

The  difference  of  longitude  between  two  places  in  the 
same  latitude,  where  the  eclipse  begins  at  sun-rise  and  ends 
at  sun-rise,  is  in  this  case  28°  30'. 

•In  latitude  29°  34'  north,  and  declination  21°  11',  the  sun  rise*  *i 
5k.  10m. 


APPENDIX  327 

Had  we  taken  the  point  g  nearer  to  c,  the  difference  of 
the  two  longitudes  would  have  been  over  30°. 

Had  y  been  taken  nearer  to  a,  the  difference  would  have 
been  less.  Had  (y)  been  at  «,  the  two  longitudes  would 
come  together. 

Lines  drawn  joining  these  longitudes  and  latitudes  will 
form  a  loop,  —  the  widest  part  will  be  not  far  from  30° 
through  c,  it  will  pass  round  in  a  curve  which  will  touch  the 
point  e,  and  will  narrow  down  to  a  point  at  (a). 

A  similar  loop  will  be  made  by  joining  the  points  of 
latitude  and  longitude  of  places  where  the  eclipse  ends  at 
sun-set  and  begins  at  sun-set. 

The  narrow  part  of  this  loop  will  begin  at  (£),  be  widest 
through  (</),  and  curve  round  and  touch  /. 

We  have  thus  far  treated  of  latitudes  and  longitudes  of 
places,  at  sun-rise,  at  noon,  and  at  sun-set.  We  will  now 
show  how  to  find  ; 

LATITUDES     AND     LONGITUDES 

Of  places  on  the  base  of  projection,  and  determine  the  Howtofind 
apparent  time    of   day  when    the    sun   will    be    centrally  latitude 


eclipsed,  or  have  the  same  apparent  right  ascension  as  the:®115111"0  of 

moon.  within       the 


We  will  assume  the  angle  ^<£R=50°,  then  MGC=4Q°. 
To  this  add  the  angle  PGC,  which  we  have  already  deter- 
mined  to  be  33°  33'  33",  therefore  the  angle  RGrP=73° 
33'  33",  GR—660  53'.  £P=26°  21'  30". 

Let  the  arc  PR  be  represented  by  x  ;  then,  by  spherical 
trigonometry,  (see  Robinson's  Geometry,  p.  191),  we  have 

cos  73°  33'  w—  cos-*~  cos-  66°  53'  cos'  26°  21<  30* 


sin.  66°  53'  sin.  26°  21'  30" 
Whence,  cos.*=sin.  Lat.=cos.  73°  33'  33"  sin.  66°  53 
sin.  26°  21'  30"+cos.66°  53'  cos.  26°  21'  30". 
cos.  73°  33'  33"     9.451832     - 
sin.  66°  53'  9.963650  cos.  9.593955 

sin.  26°  21'  30"     9.647367  cos.  9.952326 


.11557  —1.062849    .3518       —1.546281 

I 11557 



Nat.  cos.*=sin.  Lat.  27°  52'    .46737 


328  APPENDIX. 

HOW  to  find  The  natural  sine  of  any  other  point  along  this  same  small 
day  at  any  circle  c  11 Z  Qd,  can  be  determined  by  the  following  prac- 
Doiat  on  tb«  tical  equation  : 

sin.  Lat.=0.3518-|-9.611017  cos.  A. 
The  included  angle  between  GR  and  GP  is  represented 
generally  by  A.     .3518  is  a  constant  natural  number — the 
other  term  is  a  log.,  the  number  to  which  must  be  added 
to  the  constant. 

We  must  now  determine  the  angle  RPZ,  which  is  the 
angular  distance  of  the  point  R  from  the  solar  meridian, 
and  of  course  this  angle  will  give  the  apparent  time  of  day 
at  R,  at  the  moment  of  the  central  eclipse,  as  seen  from 
that  point. 

In  the  triangle  GPR  we  have  GR  66°  53',  PR  62°  8% 
and  GP  26°  21'  30",  the  three  sides  from  whence  we  find 
the  angle  GPR=93°  50'.  To  this  we  add  the  angle  GPH, 
already  determined,  (38°  26'  35"),  and  the  sum  is  the  angle 
HPR  132°  16'  35";  the  supplement  to  this  \&RPZ  47°  43' 
\  25",  the  meridian  distance  of  the  sun. 

Hence  the  apparent  time  from  noon  is  3h.  10m.  54s.,  and 
the  sun  being  east,  it  is  before  noon,  or  8h.  49m.  6s.  A.  M. 
To  find  the  time  of  conjunction  for  this  locality,  we  ob- 
serve that  nm  is  equal  to  cm  multiplied  by  the  gine  of  40°. 

That  is,  (3001)(0.6428)  =  19*9".05 

To  which  addmr,  368".5 

And  we  have  c  r  -  -      *297".55 

h.  m.   s. 
Which  divided  by  1671"  gives    1  22  31 

Add         -         -         -         -      8  49    6  A.  M. 


Time  of  tf  at  R,         -       =10  11  37  A.  M. 
Time  of  tf  at  Greenwich,        8  58  58  P.  M. 

Difference  of  meridians      10  47  21=Lon.  161°50' W. 
Thus  we  have  determined  that  in  Lot.  27°  52'  north,  and 
Lon.  161°  50'  west,  the  sun  must  have  been  centrally  eclipsed 
mi  %h.  49m.  6*.  A.  M.t  apparent  time. 


APPENDIX.  329 


And  thus  we  might  find  the  latitude  and  longitude  of 
any  other  assumed  point  on  that  line,  and  the  time  of  day 
that  the  central  eclipse  must  take  place. 

Some  points  on  this  line  are  more  easily  determined  than  Th«iatitnd« 
others.  The  point  Z  has  already  been  determined  by"£of!^ 
means  of  the  right-angled  triangle  GZH.  points  on  the 

The  latitude  of  Q  is  found  thus  :  From  GQ  66°  53'  take 
GP  26°  2 T  30",   and  we   have   (40°  31'  30")   for  the  co- than 
latitude  of  Q,  therefore   the   latitude  of  that  point  is   49° And  why 
28'  30". 

The  angle  CPQ  is  equal  to  its  opposite  angle  G  P  ff, 
(38°  26'  35"),  therefore  the  apparent  time  was  2h.  33m. 
46s.  when  the  sun  was  centrally  eclipsed  at  the  place  rep- 
resented by  Q.  To  find  the  time  of  conjunction  for  that 
locality,  we  have  the  angle  m  Gy=33°  33'  33";  ym  is  the 
natural  sine  of  33°  33'  33"  when  md  (3001.2)  is  taken  as 
radius. 

Therefore  ym=(3001.2)(.5528)=1659" 

From  this  subtract  m  r      -  368".5 

ry—    1290".5 
This  divided  by  1671"  gives  44m.  20s.  for  the  time  of 

conjunction. 

h.  m.  s.  ij,.*! 

Time  of  central  eclipse,     -     2  33  46  P.  M. 
Time  past  tf  *  44  20 

Conjunction  at  Q,  1  49  26 

Conjunction  at  Greenwich,     8  58  58 

"   Difference  of  meridians,  7    9  32=Lon.  107°  23'W. 

A  line  drawn  through  the  points  of  latitudes  and  longi- 
tudes of  c,  R,  Z,  Q,  and  d,  on  a  globe  will  fully  define 
the  central  eclipse  across  the  earth. 

To  find  the  point  on  the  northern  line  of  simple  contact 
when  it  crosses  GP,  we  simply  subtract  16°  1'  40"  from 
26°  21'  30",  and  we  have  (10°  19'  50")  for  the  co-latitude 
of  that  point,  whence  the  latitude  is  79°  40'  north. 

The  longitude  of  the  meridian  PC  we  have  found  to  be 
134°  45'  west,  therefore  the  opposit*  meridian  Pff\s  45° 


330  APPENDIX. 

38°  26' 


the  opposiu  have  83°  41'  east  longitude. 

fi<j"  °^the  Hence,  the  northern  line  of  simple  contact  passes 
found.  through  the  point  of  latitude  79°  40'  north,  and  longitude 
83°  41'  east.  Connect  this  with  the  points  of  latitude 
and  longitude  of  a,  and  of  b,  on  a  globe,  and  keep  the 
plane  of  intersection  parallel  to  the  central  plane,  and  per- 
pendicular to  OC,  and  we  shall  find  the  northern  line  of 
simple  contact  on  the  earth. 

By  extending  the  spherical  lines,  GR,  PR,  to  meet  on 
the  surface  of  the  earth,  in  the  plane  \thich  passes  through 
etf,  we  can  find  the  latitudes  and  longitudes  of  points  on 
that  line,  as  we  found  those  on  the  central  line.  Connect- 
ing these  points  on  a  globe,  will  define  the  southern  boun- 
dary of  the  eclipse  on  the  earth. 

Thus  we  have  shown  how  to  make  a  complete  delineation 
of  a  solar  eclipse  on  the  surface  of  the  earth. 

We  will  now  show  by   an  example,  a   very   accurate 
%      method  of  computing 

A    LOCAL    E  CLIPSE. 

Local  •clip-  The  greatest  eclipse,  as  seen  from  any  one  place,  will 
occur  when  the  apparent  distance  between  the  center  of 
*ne  sun  an(^  ^ie  center  °f  *-ne  moon  will  be  the  least  possi- 
ble. The  eclipse  will  commence  when  the  apparent  dis- 
tance between  the  two  centers  is  equal  to  the  sum  of  the 
two  semidiameters. 

We  can  compute  the  right  ascension  of  both  sun  and 
moon  for  any  particular  time,  and  this  would  determine 
their  distance  asunder,  provided  the  two  bodies  were  not 
displaced  by  parallax. 

To  correct  for  this,  we  must  be  able  to  determine  th« 
moon's  parallax  in  Rigid  Ascension  and  Declination. 

To  simplify  the  computation,  we  will  suppose  the  sun  to 
be  stationary  at  the  point  of  conjunction  in  Right  Ascen- 
sion, and  the  moon  to  move  with  the  difference  of  motion 
between  the  two  bodies.  We  also  conceive  the  sun  to  have 
no-  parallax,  which  we  can  do,  if  we  subtract  its  parallax 
from  that  of  the  moon. 


APPENDIX.  V 

The  value  of  the 
moon's  parallax  in  al- 
titu  le  is  represented 
by  mn,  in  the  adjoin- 
ing figure.  And  a  n 
represents  the  paral- 
lax in  right  ascension, 
and  a  m  the  parallax 
in  declination. 

Let  A  represent 
the  apparent  altitude 
of  the  moon,  and  / 
its  angular  distance 
from  the  meridian.  That  is,  t=  the  number  of  degrees  in 
the  angle  ZPm.  Also,  let  p  represent  the  reduced  hori- 
zontal parallax  of  the  moon. 

Then  mn=p  cos.^4.         (1) 

•  The  triangle  amn,  by  reason  of  its  small  magnitude,  is 
to  be  taken  as  a  plane  right  angled  triangle,  the  right  angle 
at  a. 

Then  1   :  p  cos.  A  :  :  sin.m  :  a  n.         (2) 

Let  L  represent  the  latitude  of  the  observer,  whose  zen- 
ith is  Z,  and  let  D  represent  the  declination  of  the  moon. 
Then,  by  inspecting  the  spherical  triangle  ZmP,  we  per- 
ceive the  truth  of  the  following  proportion  : 
sin.m  :  cos.X  :  :  sin.  I  :  cos.  A. 

Whence  sin.m  coa.A=cos.L  sin./.          (3) 


From  (2)  we  obtain     an—p  sin.m  cos.  A.  the 

,,r,         V     '  ,-     .  of  the  moon 

Whence  an=p  cos.Z-  sin./.  in  Right  As- 

•  —  —  —  cension,  and 

Again  am=p  cos.  A  cos.m          (4)  "»    deoiina. 

i   But  the  spherical  triangle  ZmP  gives  us 

sin.Z  —  sin.  A  sin.Z>  /e\ 

cos.m=  —  .  (5; 

cos.^4  cos.D 

The  value  of  (cos.m  COB.  A)  obtained  from  (5)  and  placed 
IQ  (4),  will  give 


cos./) 


33t  APPENDIX. 

This  equation  contains  (sm.A),  which  we  wish  to  ex 
punge,  and  the  spherical  triangle  Zmp  will  give  us 

.Z  sin./) 


cos.L  cos.  D. 

Or         sin.^=cos.*  cos.L  cos.D-^-sm.L  sin./). 
Or  sin.^4  sin./)=cofi.*cos./i  cos./)  sin./)-|-sin.Zsin.32). 
Whence         sin.Z  —  sin.  A  sin./)=sin.Z(  1  —  sin.2/))  — 

cos.*  cos  L  cos./)  sin./). 

But  1  —  sin.a/)=cos.3/).     This  value  put  in  the  second 
member,  and  both  members  divided  by  cos./),  we  shall  have 

sin.Z  —  sin.  A  sin./)        .     r         ^ 
--  =  -  =sm..L  cos./)  —  cos.*  cos.  //sin./). 

COS.// 

Comparing  this  with  (6),  we  obtain 

am=p  cos.  L  cos.  D  —  p  cos.  L  sin./)  cos.*.     (7) 
Thus  we  have  found 

Parallax  in  right  ascension  =p  cos.L  sin./. 
Par.  in  declination  —p  sin.Z  cos./)  —  p  cos.L  sin./)  cos.t. 
If  we  compare  these  expressions  with  the  solar  co-ordi- 
nates on  page  306,  we   shall  see  that  they  are  the  same. 
Hence,  the  theory  of  the  projection  agrees  with  spherical 
trigonometry. 

If  we  place       A=p  cos.Z,       /?=jt>  sin.  L  cos./),       and 
C=p  cos.L  sin./),     we  shall  have 

Parallax  in  Right  Ascension  =A  sin.*. 
And  Parallax  in  Declination  =/?  —  Coos.*. 
The  parallax  in  R.  A.  varies  as  the  sine  of  the  moon's 
meridian  distance,  and  the  parallax  in  Declination  varies 
as  the  cosine  of  the  meridian  distance  united  to  a  constant. 
The  high.     We  will  now  take  the  latitude  of  49°  20'  north,  and  lon- 
utimde  gi^U(je   io5°     west,    and    compute    the    eclipse    as    seen 
11  from  that  place.     The  result  ought  to  be  very  nearly  a 
central  eclipse. 

h.  m.     s. 
Conj.  at  Greenwich,  app.  time,  8  58  58  P.  M. 

Longitude  in  time,         -         -7 

Sun  and  moon  west  of  merid.     1   58  58  ==29°  58'  30**r-f. 


APPENDIX.  3S5 

^'s  equatorial  horizontal  parallax,  54'  32V.6 
j^'s  horizontal  parallax,  8".5 

54'  24".  1 

Reduction  for  latitude  49°  -  7".l 


54 

'  17 

"  =3257"=;). 

P 

- 

- 

3 

,512818 

P 

3.512818 

sin, 

L 

49° 

20' 

9, 

879963 

cos 

.L 

. 

9 

.814019 

cos.D 

21 

0  33'  32" 

9 

.968503 

sin 

.1)  • 

-   9 

.565187 

,  — 

cos 

.*29° 

58' 

30"  9 

.937640 

B 

2297 

6 

3 

.361284 

675 

6 

. 

„ 

„ 

2 

.829664 

1622"=parallax  in  declination. 
A=         3.326837 
sin.*=         9.698641          (7=2.892024 

Parallax  in  R.  A.  1060".4     3.025478 

We  will  now  compute  the  parallax  in  right  ascension  and 
declination  at  the  expiration  of  half  hour  intervals  after 
this  time,  as  follows  : 

At  time  of  conjunction  in  this  locality,  /=29°  58'  30" 

30  minutes  interval,  -f-  7°  30' 

Q)'s  motion  from  ^  during  this  interval,  —         14'  23".6 

First  half  hour  after  conjunction,       -      /=37°  14'    6" 
Motion  from  the  meridian  in  30m.  7°  15'  36" 


One  hour  after  conjunction,  /=44°  29'  42" 

Motion  from  the  meridian  in  30m.  7°  15'  36" 


One  hour  and  thirty  minutes  after  conj.        t=5l°  45'  18 
Two  hours  after  conjunction,  *=59°    0'  54" 


Parallax  in  Right 
found  as  follows  : 
1st  half  hour. 
A  3.326837     A 
sin.*  9.781814 

Ascension, 

3.326837 
9.845632 

in  half  hour 

3d. 
A  3.326837 
9.895063 

intervals,  is 

Practical 
computatioa 
4th.              of  parallHX. 

A  3.326837 
9.933140 

At^    3.108651 
1060."4   1284".3 

3.172469 
1487".4 

3.221900 
1666".7 

3.259977 
1819". 

APPENDIX. 

The  right  ascension  of  the  moon  is  east  of  the  sun  at 
(f         0    £h.  af.  863"6  Ih.  1727"2  l£h.  2590"8  2h.3454"4 
)par.  1060"4        1284"3         1487"4  1666"7        1819" 


Q)  N.  1060"4 
cos.Q)D..93 

W.  420"7 
.93 

E.  239"8 
.93 

924"! 
.93 

1635"4 
.93 

3181 

1262 

7194 

2782.3 

4906.2 

95436 

3786 

21582 

83469 

147186 

986.17       391.22      223.014  862.513     1529.922 

<3)W.of  Q.  Q)  W.  of  Q.  Q)  E.  of  Q.  Q)E.  of  Q.  Q)E.  ^. 
Thus  are  the  apparent  distances  in  Right  Ascension  re- 
duced to  the  arc  of  a  great  circle. 

For  the  parallax  in  Declination,  at  these  several  inter- 
vals, we  operate  thus : 

(72.892024   (72.892024   (72.892024   (72.892024 
cos.*      9.901001       9.853252       9.791717      9.711625 


X 

2.793025 

2.745276 

2.683741 

2.603649 

620"9 

556"2 

482"8 

401"5 

B    2297"6 

2297"6 

2297.6 

2297"6 

Q)  par.  in 
Q)N.of( 

D.  1676"7 
P    1565"7 

1741  "4 
1796"4 

1814"8 

2027"! 

1896"!     . 
2257"8 

Q)app.  S.of^lll"        N.     65"       N.    212"3  361"7 

At  conjunction  the  Q)  was  north  of  ^,  1335",  but  the 
parallax  in  declination  was  then  1622"  southward.  Hence, 
the  apparent  declination  must  have  been  287"  south. 

Now  to  determine  the  beginning  and  end  of  the  eclipse, 
and  other  circumstances,  we  must  resort  to  a  partial  pro- 
jection, as  follows : 

Let  S  represent  the  center  of  the  sun  at  the  time  of  con- 
junction in  right  ascension,  apparent  time. 

The  true  right  ascension  of  the  moon  is  then  the  same 
as  that  of  the  sun.  But  the  parallax  in  R.  A.  we  have 
found  to  be  C86"2,  westward  of  course,  because  the  moop 
is  west  of  the  meridian. 


APPENDIX. 

*  Draw  the  horizontal   £a=986"2,  and  from  a  make  am= 
287",  and  ra  is  the  apparent  place  of  the  moon  at  the  time  of  ,caie 
conjunction. 


Projection 


: 


One  half  hour  afterwards  the  apparent  place  of  the  moon 
was  391"  west  of  the  ^,  and  83"  south. 

We  therefore  take  £a'=391"2,  and  «7'm'=M  1",  andm'  is 
the  apparent  place  of  the  Q)  30  minutes  after  ^,  and  mm' 
is  the  apparent  motion  of  the  Q)  during  the  half  hour. 

At  the  expiration  of  the  next  half  hour,  the  apparent 
place  of  the  Q)  was  223"  east  of  the  ^  at  b,  and  55"  north 
of  that  point  at  m".  In  the  same  manner  we  determine  the 
points  ma  and  w4. 

The  position  of  the  center  of  the  sun  is  stationary  at  89 
but  the  apparent  places  of  the  moon,  are  at  m,  m',  m",  ma, 
m4,  during  the  interval  of  two  hours. 

By  merely  inspecting  this  figure  we  perceive  that  about 
46  minutes  after  ^  the  two  centers  will  be  nearest  to  each 
other,  and  they  will  not  be  3"  asunder.  That  is,  at  (Ih, 
69m. -{-46m. ),  or  2h.  45m.  apparent  time,  the  sun  will  be 
centrally  eclipsed. 

By  the  figure,  we  can  determine  the  distance  be- 
tween S  and  m",  S  and  m',  &c.,  and  their  rate  of  approach 
and  departure,  and  thence  we  can  determine  the  time  of 
beginning  and  ending,  the  time  of  forming  of  the  ring,  <fec. 

The  semidiameter  of  the  sun,  at  that  time,  was  949",  ing  and  «nd, 
that  of  the  moon,  at  the  altitude  it  then  had,  was  not  one 
second  from  900".  The  sum  of  the  two  is  1849",  and  dif- 
ference  49",  and  when  the  moon  arrives  within  49"  of  the 
center  of  the  sun,  the  ring  will  form,  and  when  it  recedes 
tb  the  distance  of  49",  the  ring  will  break. 

The  distance  from  S  to  m,  is  the  hypotenuse  of  the  right 
tngled  triangle,  whose  sides  are  986"2  and  287",  therefore 


w  to  find 
the       begin- 


336  APPENDIX. 

£m=1027".     This  was  passed  over  in  46m.  at  the  rate  ot 
22"3  per  minute. 

Hence,  from   1849  take  1027,  and  we  have  822",  which 

divided  by  22"3,  gives  36m. 

h.  m.   s. 
From  tf  -         -         -         1  58  58 

Rsteofap.  Subtract     -  36  40 

parent      mo-  -      - 

•ion  of  the          Apparent  time  of  beginning,       -          1   22  18 
tchpw   dii.          Time  of  nearest  approach,       -         -     2  44  36 

ooTorcd. 

Assuming  22"3  per  minute  for  the  apparent  motion  of 
the  moon,  and  49"  for  the  excess  of  the  sun's  semidiameter, 
Dnration  twice  49"  divided  by  22"3  will  give  4m.  23s.  for  the  con- 
ofih«ring.    tinuance  of  the  ring. 

The  rate  of  the  Q)'s  apparent  motion  between  tn3  and  m4 
is  22"5  per  minute,  and  is  on  the  increase. 

The  line  of  apparent  motion  is  a  little  curved,  becoming 
less  inclined  to  the  horizontal. 

"JVith  this  exposition  we  think  no  one  who  pays  atten 
tion,  can  fail  to  perceive  the  rationale  of  the  computation  of 
a  general  and  local  solar  eclipse. 

Thus  having  the  apparent  distances  between  m  and  S, 
m'  and  S,  <fec.  at  definite  times,  and  having  the  apparent 
rate  of  motion  of  the  moon  over  the  face  of  the  sun,  we  can 
readily  determine  the  time  within  a  few  seconds,  when  the 
eclipse  will  begin  or  end,  or  attain  any  definite  phase. 


COMPUTATIONS 

TO   FIND    THE    TIMES    WHEN    THE    MOON,    STAR,    OR    PLANET, 
WILL    RISE,   OR    SET,  ON    ANY    GIVEN    DAT. 

SEVERAL  teachers  have  requested  the  author  of  this  work, 
to  insert  a  method  of  computing  the  time  when  the  moon 
will  rise  or  set  on  any  particular  day,  as  seen  from  any 
particular  place. 

We  now  comply  with  this  reasonable  request,  hoping  to 
be  both  brief  and  clear. 

Let  the  object  be  the  moon,  planet,  or  fixed  star,  the 


APPENDIX.  387 

principles  on  which  the  computation  is  made,  are  the  same, 
And  for  the  sake  of  perspicuity,  we  will  commence  with  a 
fixed  star. 

The  following  elements  must  be  obtained  for  the  com-      Ei«m«»M 
putation:  tob«M.d. 

1.  The  latitude  and  longitude  of  the  place. 

2.  The  Right  Ascension  and  Declination  of  the  heavenly     ." '«    , 
body,  (moon  or  star,)  at  the  time  sought,  or  as  near  the 

time  sought,  as  possible. 

3.  The  Right  Ascension  of  the  sun  at  the  same  time. 
Preparatory  to  the  solution  of  this  general  problem,  we 

.  , .  .       General  e»» 

must  remember,  that  when  the  sun  passes  any  meridian,  itpianation  of 
is  then  and  there  apparent  noon  —  and  at  1  o'clock,  P.  M.  £e|>««l  P"* 
apparent  time,  the  right  ascension  of  the  meridian  is  one*'f 
hour  greater  than  the  right  ascension  of  the  sun.      When  a 
star  or  planet  is  on  the  meridian,  the  right  ascension  of  the 
meridian  is  the  same  as  the  right  ascension  oj  that  star. 

If  we  subtract  the  right  ascension  of  the  sun  from  the 
right  ascension  of  a  star,  the  remainder  is  the  apparent 
time  for  that  star  to  pass  the  meridian.  That  is, 

Apparent  time  -^  on  merid.=R.  A.  -^ — R.  A.  ^ 

The  time  from  the  meridian  to  the  horizon,  or  the  time 
from  the  horizon  to  the  meridian,  is  called  the  semi-diurnal nalare* 
arc,  as  we  have  before  explained.  Corresponding  to  cer- 
tain latitudes  of  the  observer,  and  degrees  of  declination 
of  the  heavenly  body,  it  is  to  be  found  in  a  table  on  page 
331,  or  it  can  be  computed  in  each  case,  independently  of 
the  table,  as  taught  on  page  236. 

EXAMPLES. 

1.  What  time  will  the  fixed  star  Sirius  rise,  pass  the  me- 
ridian, and  set,  on  the  20th  of  January,  1858,  as  seen  from 
Latitude  40°  N.  and  Longitude  75°  W? 

In  Table  II,  we  find  the  right  ascension  and  declination 
of  Sirius,  with  its  annual  variations.     From  1846  to  1858, 
is  twelve  years.     Hence 
22 


338  APPENDIX. 

R.  A.  h.    m.  s.  Declination. 

6  38  21.884         16°  30' 32"83  8. 
Variation  12y  +31.722  —53.80 

^-'s  position,  1858,       6  38  53.605         16°  29'  39"03 
Tb«  reason      f^Q  rjght  ascension  of  the  sun  is  zero,  on  or  about  the 
fa  Uw'oplra1!  20th  of  March  in  each  year,  and  it  increases  about  2h.  each 
tfc«.  month,  therefore,  on  the  20th  of  January,  it  cannot  be  far 

from  20  hours.  This  subtracted  mentally  from  the  right 
ascension  of  the  star,  shows  us  that  the  star  must  p'ass  all 
meridians  on  that  day  when  the  local  time  at  each  place, 
cannot  be  far  from  lOh.  30m. 

When  it  passes  the  meridian  of  Lon.  75°  west,  the  local 
time  in  that  longitude  must  be  near  lOh.  30m.  and  the  time 
at  Greenwich  15h.  30m. 

Howtoob-  Therefore,  to  obtain  a  result  nearly  accurate,  we  mus* 
tain  the  snn»«j]ave  ^Q  j-jg^  ascension  of  the  sun  corresponding  to  Jan- 
•ion.  u(jry  20th,  15h.  30m.  of  Greenwich  time,  which  is  8£h, 

previous  to  noon  of  the  21st  of  January. 

The  right  ascension  of  the  sun  is  given  in  the  Nautical 
Almanacs  for  the  noon  of  each  day  at  Greenwich,  and  the 
hourly  variation. 

h.  m.   s. 
1858,  Jan.  21,  Q's  R.  A.      20  13  53.9 

Variation  Ih.  10s.54X8£          —1  29.6 

20  12  24.3 

h.  ID      8. 

Right  Ascension  of  Sirius,     -        -          6  38  53.6 
U  Right  Ascension,  -         -     201224.3 

Sirius  passes  meridian,  (apparent  time,)  10  26  29.3 
Equation  of  time,  (Nautical  Almanac,)  +11  34.6 

Star  passes  (Ion.  75)  mean  time,     -       10  38    3.9  P.  M. 

Now  to  find  the  time  this  star  will  rise  and  set,  we  must 
apply  the  semi-diurnal  arc,  so  called,  which  can  be  found 
nearly,  in  the  table  on  page  321,  corresponding  to  Lat.  40° 
N.  and  Dec.  16°  29'.  The  table  will  give  us  6h.  57m.  30s., 
and  this  would  be  the  interval  sought,  provided  the  decii- 


APPENDIX.  339 

nation  was  north.     It  being  south,  we  must  subtract  this 
sum  from  12h.     Hence  the  arc  sought  is  5h.  2m.  30s. 

h.     m.  Approximate 

Now  the  star  passes  the  mer.  mean  time,    10  38    4 
Subtract  the  semi-diurnal  arc,         -          5    2  30 


Star  rises  (mean  time)  P.  V  6  35  34 


(Add  semi-diurnal  arc.)  Star  sets,  A.  &L.     o  4U 
next  morning. 

The  time  the  star  passes  the  meridian  is  tolerably  accu-    Corrections 

I  to  b«  — • «- 


rate,  but  the  times  of  rising  and  setting  require  correction  and  for  whau 

for  refraction.     That  cause  would  increase  the  semi-diurnal 

arc  about  2  minutes;  and  in  addition  to  this,  we  must  con- 

sider that  the  sun  changes  its  right  ascension  during  the 

6h.  2m.  that  the  star  requires  to  pass  from  the  meridian  to 

the  horizon. 

The  change  in  the  sun's  right  ascension  for  one  hour  is 
10s.  54;  in  5h.  2m.  the  change  will  be  53s. 

That  is,  when  the  star  actually  rises,  the  sun's  right  as- 
cension is  53s.  less  than  when  it  passes  the  meridian,  and 
it  is  53s.  greater  when  the  star  sets.  Hence  a  slight  cor- 
rection is  necessary.  Or  we  may  view  this  problem  from 
another  stand  point. 

When  the  star  is  rising  it  is  in  the  eastern  horizon,  and       Anotht* 

h.  m.    8.  mode  of  com. 


Its  Right  Ascension  is      -         -         -          6  38  53.6       JT|f  of  *£ 
Semi-diurnal  arc  -f-  refraction     -         —  5    4  30  sing  »nd  set. 

_  ting    of   tb* 

Diff.  =  Right  Ascension  of  meridian,          1  34  23.6        *tar* 
Sun's  Right  Ascension  at  this  time,      20  11  31.3 

Star  rises,  apparent  time,         -  5  22  52.3 

Equation  of  time,  add       ...     +11  34.6 

Star  rises  (mean  time,)  -        -  5  34  27 

When  the  star  sets,  it  is  in  the  western  horizon,  and  the 
Right  Ascension  of  the  meridian  is  greater  (or  eastward,) 
than  that  of  the  star. 


940 


APPENDIX. 

h.  m.    s. 

:'s  Right  Ascension,         -         -  6  38  63.6 

;mi -diurnal  arc  -J-  Refraction,         -       -}-5     4  30 


Right  Ascension  of  the  meridian,         -     1  1   43  23.6 
Right  Ascension  of  sun,  20   1317.3 

Star  sets  (apparent  time,)          -         -      15  30    6.3 
Equation  of  time,      ...  -j-  11  34.6 

Star  sets  (mean  time,)         -         -         -     15  41  41 
Or,  3h.  41m.  lls.  A.  M.  on  the  21st  of  January. 
Practical     ^n  solving  problems  of  this  kind,  practical  men  make  no 
not  attempt  at  accuracy,  as  the  element  of  refraction  is  very 


unable.    uncertajn  jn  j^g  results,  and  VQVJ  often  the  stars  cannot  be 
seen  at  all,  when  near  the  horizon. 

2.  Giving  the  operator  the  use  of  a  Nautical  Almanac  for 
1858,  we  require  him  to  determine  what  time  of  day  tie  planet 
Jl/ars  will  pass  the  meridian  of  Boston,  (Lat.  42°  22'  N.,  and 
Lon.  4h.  44m.  W.,)  on  the  %d  day  of  July.  Also  required 
the  time  it  will  rise  and  set. 

On  the  2d  of  July,  1858,  the  Right  Ascension  of  Mara 
is  14h.  53m.  20s.,  and  Declination  19°  1'  south,  and  these 
elements  may  be  taken  as  invariable  during  that  day. 
The  times      The  Right  Ascension  of  the  sun,  on  the  2d  of  July  of 
"^'any  year,  is  not  far  from  6h.  44m.,  and  this  mentally  sub- 
rising     attracted  from  14h.  53m.,  leaves  8h.  9m.,  to  which  add  the 
•  setting  of  the  longitude  of  Boston  in  time,  4h.  44m.,  and  we  have  Ifch. 
53m-  °f  Greenwich  time,  to  which  the  sun's  Right  Ascen- 

sion must  correspond. 

h.  m.    s. 

R.  A.  of  sun,  July  2d,  '58,  at  noon,  Gr.  6  44  33.9 
Variation  per  hour,  10s.3X(12.53)  -)-2  12.1 

R.  A.  of  sun  when  moon  is  on  merid.      6  46  46 
From  Right  Ascension  of  Planet,  14  53  20 

Subtract  Right  Ascension  of  sun,         6  46  46 

Mars  passes  meridian,  (app.  time,)  8    6  34  P.  M. 

Equation  of  time,  (add)       -         -          +343 


Mars  on  meridian,  (mean  time,)  8  10  17 


APPENDIX.  341 

For  latitude  42°  22',  and  Declination  19°  1',  the  semi- 
diurnal  arc  is  7h.  13m.,  or  4h.  47m.  In  this  case  it  is  4h. 
47m.,  the  Latitude  being  north  and  Declination  south. 

h.   m.    s. 

Hence,  from  and  to          -        -  8  10  17 

Subtract  and  add        -        -        -        4  47 


Mars  rises,  (mean  time,)         -        -     3  23  17  P.  M. 
Mars  sets,  "  -        -       12  57  17  P.  M. 

Or  Oh.  57m.  17s.  on  the  morning  of  the  3d  of  July. 

The  planet  rises  when  the  sun  is  up,  and  r  f  course  it 
will  be  invisible. 

Here  neither  refraction  nor  the  change  irt  the  sun's  B. 
A.  are  taken  into  account,  and  it  is  not  important  that  they 
should  be, — however,  for  the  improvement  of  the  student, 
we  will  compute  the  time  the  planet  sets,  increasing  the 
semi-diurnal  arc  2m.  for  refraction,  and  50s.  for  the  in- 
crease in  the  sun's  Right  Ascension. 

h.  m.    B. 

Bight  Ascension  of  Mars,     -     14  53  20 
Semi-diurnal  arc  -|-2m.     •  4  49 


Bight  Ascension  of  meridian^     19  42  20 
B.  A.  of  sun  at  this  time,  6  47  26 


Mars  sets,  (app.  time,)         -      12  54  54 
Equation  of  time,  -(-        -        -     3  44 

Mars  sets,         -        -      :^-fl       0  57  38  A.  M.  July  3d. 

We  are  now  prepared  to  apply  these  principles  to  the 
moon. 

Several  years  ago,  when  only  the  longitudes  and  latitudes  Of  this 
of  the  moon  were  given  in  the  Nautical  Almanac,  the  rising  blem 
and  setting  of  the  moon  was  a  problem  of  some  complexity, 
but  recently  the  right  ascensions  and  declinations  of  the 
moon  are  computed  and  written  down,  corresponding  to 
every  hour,  in  both  the  English  and  American  Nautical 


APPENDIX. 

Almanacs,  and  every  student  of  Astronomy  should  hare  a 
copy.* 

Whe*i  the  moon  changes,  as  it  is  called,  it  sets  about  the  time, 
or  a  little  before  the  sun. 

When  the  moon  fulls,  it  rises  about  the  time  of  sun-set. 
We  compute  the  successive  times  the  moon  sets,  com- 
mencing with  new  moon  and  closing  with  full  moon.  Then 
commence  computing  for  moon-rise,  from  full  moon  to  ne^w 
moon. 

FOR      EXAMPLE. 

Having  a  Nautical  Almanac  for  1857  before  us,  we  find 
that  the  moon  changes  or  passes  the  sun,  January  25th, 
5h.  49m.  P.  M.,  mean  time  at  Cincinnati.  It  will  go  down 
invisible  in  the  blaze  of  sun-light  that  day,  a  little  before 
or  a  little  after  the  sun,  according  to  the  relative  declina- 
tion^ of  the  two  bodies.  The  exact  time,  for  the  day  of 
change  is  never  computed,  unless  it  be  in  connection  with 
an  eclipse. 

A  definite      What  time  will  the  moon  set,  January  26th,   1 857,  as  seen 
««npie.       from  Cincinnati,  Latitude  39°  6'  N.,  Longitude,  in  time,  5h. 
37m.  west  of  Greenwich? 

4*  •' 

SOLUTION. 

^Preparation.  On  the  26th  of  January,  in  Lat.  39°  N,  the  sun  sets  at 
5h.  10m.  mean  time,  and  the  moon  I  judge  will  set  Ih,  af- 
terwards, at  6h.  10m.  To  this  add  the  longitude,  5h.  37m. 
and  the  Greenwich  time  is  thus  determined  to  be  1  Ih.  47m. 

In  the  Nautical  Almanac  we  find  that  the  right  ascen- 
s'on  °^  *^e  moon  at  ^is  time  is  21h.  35m.  2s.,  and  decli- 
nation  18°  21'  15"  S. 

The  semi-diurnal  arc  corresponding  to  Lat.  39°  6'  N., 
and  declination  18°  21'  S.,  is  7h.  3m.  from  12h.,  or  4h.  57m 
Hence,  the  computation  is  as  follows : 

*  Nautical  Almanacs  are  made  at  public  expense,  and  sold  very 
cheap  for  the  promotion  of  science.  The  price  of  a  single  copy  is  from 
50  cents  to  $1 ,  barely  enough  to  cover  the  cost  of  printing  and  paper. 


APPENDIX.  S43 


Q)'s  R.  A.  Jan.  26  (at  6  10  P.  M.)  21   as     2 

Semi-diurnal  arc,         add  4  57 


Right  A.  of  meridian,  26  32     2 

R.  A.  of  the      ,  sub.  20  37  44 


Q)  sets  (apparent  time),  Cincinnati,  5  54  18  P.  M. 

Equation  of  time,  Nautical  Almanac,      -|-  13     1 

Q)  sets,  mean  time,  6     7  19 

This  computation  makes  no  allowance  for  refraction,  or   Allowances 
for  parallax.     Within  the  latitudes  of  40°  on  each  side  of  *°  *•  made 

*  for  refraction 

the  equator,  the  moon  is  generally  kept  above  the  horizon  ana  parallax. 
about  two  minutes  longer  by  refraction,  and  will  set  about 
four  minutes  earlier  in  consequence  of  parallax.  The 
effect  of  the  two  causes  combined  make  the  moon  set  two 
minutes,  or  more,  sooner  than  is  given  by  the  preceding 
result.  Therefore,  if  we  were  making  an  Almanac  for 
1857,  for  the  locality  of  Cincinnati,  we  would  record  the 
setting  of  the  moon  January  26th,  at  6h.  5m. 

Having  the  Nautical  Almanac  before  us,  and  knowing 
the  moon  to  be  near  her  perigee,  and  her  south  declination  tinned 
decreasing,  we  know  that  the  moon  must  set  on  the  eve- 
ning of  the  27th,  more  than  one  hour  later  —  we  judge 
about  7  15.  This  will  make  the  Greenwich  time  near 
13h.  Corresponding  to  which  time,  we  find  the  moon's 
right  ascension  and  declination,  as  before,  and  compute  the 

time  it  sets. 

h.  m.    s. 

Thus,  Q)'s  R.  A.—  N.  A.,    -  22  31     0 

(Q)'s  Dec.  12°  19'  S.)  semi-diurnal  arc,  5190 

R.  A.  of  Meridian,  -  -     27  50     0 

R.  A.  of  Sun,  N.  A.    (Sub.)          -        20  42    0 

Q)  sets  (apparent  time),         -         -  780 

Equation  of  time,  N.  A.      -         -         -         13  12 


Q)  sets,  mean  time,  Cincinnati,         -       7  21    12 
And  thus  we  go  on  from  day  to  day. 


344  APPENDIX. 

TO     COMPUTE      THE     TIME     THE     MOON     RISKS. 

Anexanpi*      On  the  8th  of  February,  between  Band  7  P.  M.,  Cincin- 
**T*n-          nati  time,  the  moon  fulls.     What  time  will  it  rise  on  the 
9th? 

We  judge  that  it  will  rise  about  one  hour  after  sunset, 
or  about  6h.  13m.     Adding  5  37  we  obtain  llh.  60m.  for 

the  Greenwich  time. 

h.  m.    8. 

Th«  com-      Q)'s  R.  A.  at  that  time,  Nautical  Almanac,  10  24  25 

Semi-diurnal  arc,         (sub.)  6  43  20 


Right  ascension  of  meridian,  341     5 

R.  A.  at  that  time,  Nautical  Almanac,  21  30  39 


)  rises,  apparent  time,  6  10  26 

Equation  of  time,  Nautical  Almanac,          -|-  14  31 


»  mean  time,  (unconnected), 
Correction  for  R.  and  P. 

)  rises,  mean  time,  corrected,  6  27 


•  The  critical  student  will  be  desirous  to  learn  how  to  compute  the 
precise  effect  of  parallax  and  refraction  on  a  heavenly  body,  just  at 
the  point  of  rising  or  setting.  To  show  this  we  take  the  following: 
example  : 

In  latitude  60°  North,  when  the  moon's  declination  is  20°  North, 
horizontal  parallax  58',  and  refraction  34',  what  is  the  semi-diurnal 
arc,  refraction  and  parallax  being  duly  allowed  for  ? 

CONSIDERATION.  —  The  moon  is  depressed  58'  by  parallax,  and  de- 
rated 34'  by  refraction  ;  therefore,  the  moon,  in  the  horizon,  is  depressed 
24'  and  will  set  when  a  star,  or  the  sun,  at  the  same  point,  as  seen 
from  the  center  of  the  moon,  would  be  24'  above  the  horizon. 

Therefore,  to  find  the  true  semi-diurnal  arc  for  the  moon,  we  con- 
ceive a  star  at  the  altitude  of  24',  latitude  60°,  and  polar  distance,  and 
compute  the  polar  angle,  as  explained  on  page  251. 

This  gives  the  semi-diurnal  arc  for  the  moon,  in  this  example, 
8h.  31m.  58s. 

But  the  semi-diurnal  arc,  without  refraction  or  parallax,  is  8h.  36m. 
20s.  Hence,  the  effect  of  parallax  and  refraction  in  this  example  is 
4m.  22s.  That  is,  the  moon  will  set  4m.  22s.  earlier,  or  rise  4ci  22*. 
later  than  it  would,  unaffected  by  refraction  and  parallax. 

And  thus  we  could  compute  exactly  in  any  other  example. 


APPENDIX.  34* 

On  the  10th,  the  moon  will  rise  about  one  hour  later, 
or  at  7h.  30m.,  to  which  add  the  longitude  in  time,  5h. 
37m.,  and  the  sum  is  )3h.,  Greenwich  time. 

At  that  time  the  moon's  right  ascension,  by  the  Nau- 
tical Almanac,  was  llh.  11m.  34s.  and  its  declination 

7°  11'30"N. 

h.    in.     s. 
Q)'s  R.  A.  (at  13h.,  Greenwich  time),   11   11  36 

Semi-diurnal  arc,         (sub.)          -  6  29 


Right  A.  of  Meridian.  -  4  42  36 

^'s  11.  A.  (Nautical  Almanac),       -     21   38  44 

Q)  rises,  apparent  time,         -  7     3  52 

Equation  of  time,  Nautical  Almanac,        14  31 
Correction  for  Ref.  and  P.    -                      2 


Q)  rises,  mean  time,         -         -         -       7   1 9  23 

From  latitude  39°  North,  and  Q)'s  declination,  13°  6' 
North,  we  find  the  semi-diurnal  arc,  to  be  6h.  43m.  20s.  as 
above. 

Thus  we  may  go  on  from  day  to  day. 

The  labor  of  making  these  calculations  is  not  so  great  The  labor  t» 
as  it  appears  to  be  in  these  pages.  b^oiTe^fc! 

Here  we  are  teaching  the  pupil,  and  were  compelled  to  miliai. 
write  out  all  our  thoughts.     In  actual  calculation  we  write 
out  only  the  figures,  and  do  not  carry  it  to  seconds  for 
common  almanacs. 


ARGUMENTS       FOR     EQUATING      THE      MOON*S      LON- 
GITU  DE. 

In  the  lunar  table,  we  find  20  Arguments  for  the  moon's    Gene«i«. 
longitude,  and  we  have  been  requested  to  explain  them,  or  Plan»tioni- 
show  to  what  1,  2,  3,  4,  <fec.  correspond. 

The  sun  is  the  only  body  that  sensibly  disturbs  the  mutual 
motions  of  the  earth  and  moon  ;  and  all  motions,  whether 
they  be  of  the  earth  or  moon,  we  attribute  to  the  moon 
alone. 


546  APPENDIX. 

The  mean  lunar  motion  is  variable,  according  to  the 

rS 
variable  value  of  the  expression  --j  (see  Art.  179).     But 

it  is  not  necessary  to  explain  all  this  over  again,  The 
variations  of  a,  correspond  to  the  sun's  anomaly,  abbrevi- 
ated thus :  QAn. 

The  variations  of  r,  correspond  to  the  moon's  anomaly, 
abbreviated  thus :  Q>An. 

The  variations  of  ^An.  and  §)An.,  combined  with  all 
the  possible  positions  of  the  sun,  moon,  and  moon's  node, 
Till  produce  variations  in  the  moon's  motion. 

For  the  first  ten  Arguments  the  circle  of  360°  is  sup- 
- posed  to  be  divided  into  10,000  equal  parts;  and  from  10 
to  Arg.  20  it  is  divided  into  1,000  equal  parts. 

The  first  Argument  corresponds  to  the  annual  equation, 
caused  by  the  sun's  variable  distance. 

The  sun  moves  from  his  perigee  to  perigee  again,  in 
365  days  13h.  Dividing  10,000  by  365d.  13h.,  gives  us 
27.36  for  one  day  —  the  mean  motion  of  the  sun's  anomaly. 

The  symbol  (Q) — ®)>  indicates  that  the  sun's  mean 
motion  must  be  subtracted  from  that  of  the  moon. 

"With  these  explanations,  we  suppose  that  all  the  follow- 
ing indications  will  be  understood: 

ARGUMENTS     FOB     LONGITUDE. 

Motion  in  24  hours 

Arg.   \=^An.  =     27.36 

2=2(Q)  —  Q)— ^An.  =  650.08 

3=Arg.2+(j)An.-\-QAn.        =1040.35 
4=Arff.2—(3)An.  =  287.17 

=  335.55 
=  372.03 
=  57.68 
=  390.27 

9=^An.— (^Perigee  =     24.24 

10=2(3— Q— 3^.  )+QAn.=     70.17 
1 1  —Evection—  Per.— Node  =     31.28 

12=2(3— &)+9An.  =     70.552 


APPENDIX.  347 

*  =  34.19 

14=3.4r0r.l3—  gfrom  QjAbcfe  =  99.15 

\b=Evecti&n—ZPer.—%tNode  —  30.54 

16=Q)'s  motion  from  Node  =  36.74 

17=Z<m.Q)+2Z0w.^  =  42.07 

\%=Evection  —  ZLon.Q  =  25.994  nearly, 
19=Motion  of  Q)'s 
20=Motion  of     )'s 


ARGUMENTS     FOR     LATITUDE. 

Arg.  i.  The  first  Arg.  is,  of  course,  moon's  distance  from 
her  node. 


Arg.  n.  The  second  Arg.  is  the  double  distance  of  moon 
from  sun,  minus  the  moon's  distance  from  her  node. 


That  is,  2#=2Q)—  2^         ^=Q)+  node. 

2d  Arg.  =2^—^=3—  2^—  node. 
Moon's  motion  in  one  day,  13°   10'    35" 

—  Double  sun's  motion,  59'  8"     1     58     16 


11     12     19 

Minus  motion  from  node,  3     11 


One  day's  motion  for  Arg.  n.      11°     9'      8y 

Arg.  in.  Is  the  moon's  longitude. 

N.  B. — This  circumstance  would  not  affect  the  latitude* 
if  the  earth  were  a  perfect  sphere.  The  same  remark  will 
apply  to  the  next  argument — the  20th  of  longitude. 

Arg.  iv.  Is  Arg.  20  of  longitude. 

N.  B.  Whatever  affects,  i.  e.  accelerates  or  retards  the 
moon's  longitude,  affects  its  latitude  proportionally,  unlesg 
the  moon  be  90°  from  her  node. 


348  APPENDIX. 

The  following  elements  for  a  solar  eclipse,  in  the  year 
1858,  will  give  the  student  an  example  independent  of  our 
comments  and  illustrations. 


An  Annular  Eclipse  of  the  SUN,  March  14-15,  1858,  visible 
(as  a  partial  one),  at  Greenwich. 


Elements 
from  theNan- 
tical  Alma. 


ELEMENTS. 


h.  m.       s. 

Greenwich  Mean  Time  of  tf  in  Right  As- 
cension, March  15,  ------         044    7.6 

Sun  and  Moon's  Right  Ascension     -     -       234025.14 

O        /          If 

Moon's  Declination S.  1  24  21.8 

Sun's  Declination S.  2    7  15.0 

Moon's  Horary  Motion  in  R.  A.    -     -  30  16.4 

Sun's  Horary  Motion  in  R.  A.      -     -  2  17.1 

Moon's  Horary  Motion  in  Declination     N.  16  30.3 

Sun's  Horary  Motion  in  Declination     N.  59.2 

Moon's  Equatorial  Horizontal  Parallax  58  15.2 

Sun's  Equatorial  Horizontal  Parallax     -  8.6 

Moon's  true  Semidiameter  -     -     -     -  15  54.6 

Sun's  true  Semidiameter     ...     -  16    6.5 


General  re.  Begins  on  the  Earth  generally  March  14,    21h. 
nhs  from  the      3lm. 6,  Mean  Time  at  Greenwich,  in  Longi- 
tude 50°  47' W.  of  Greenwich,  and  Latitude     4°  26'S. 

Central  Eclipse  begins  generally  March  14,  22h 
42m.  1,  in  Longitude  67°  50'  W.  of  Greenwich, 
and  Latitude  11°  19'N. 

Central  Eclipse  at  noon,  March  15,  Oh  44m.  1, 
in  Longitude  8°45'W.  of  Greenwich,  and 
Latitude  45°  44'N. 


APPENDIX. 


349 


Central  Eclipse  ends  generally  March  15,  Ih 
28m.  1,  in  Longitude  64°  40'  E.  of  Greenwich, 
and  Latitude  69°  19'N. 

Ends  on  the  Earth  generally  March  15,  2h  38m. 6, 
in  Longitude  49°  44'  E.  of  Greenwich,  and 
Latitude  53°  46'N. 


From  the  preceding  elements  the  English  Astronomers 
have  denned  the  line  of  the  central  eclipse,  by  the  follow- 
ing table  of  latitudes  and  longitudes. 

The  student,  if  he  operates  as  taught  in  the  preceding  N0twoop«- 
pages,  will  tind  nearly  the  same  line,  but  he  will  not  findrator»  wi" 
the  same  points,  unless  he  works  from  precisely  the  same 


triangles 


but  that  is  by  no  means  probable. 


Line  of  Central  and  Annular  Eclipse. 

Longitude. 

Latitude. 

Longitude. 

Latitude. 

o         / 

0             / 

0             / 

0             / 

67     50  W. 

11     19N. 

18     25W. 

35     25N. 

56     34 

11     40 

8     20 

46       7 

51      12 

12     30 

1      18W. 

51     49 

43     54 

14     33 

9     17E. 

58       2 

37     10 

17     44 

23     10 

63     18 

27     25 

25     24 

46     40 

67     56 

23       9W. 

29     57  N. 

64     40  E. 

69     19N. 

The  Southern  line  of  simple  contact  will   pass  through 
the  following  points  of  latitude  and  longitude  : 


Longitude. 

62  51 W. 

53  29 

38  20 

30  43 

15  53 

10  9 

4  49 


Latitude. 

Longitude. 

Latitude. 

o         / 

o         / 

0             / 

23     50  S. 

1       9E. 

6     37N. 

23     45 

6     29 

12     42 

21      44 

19     22 

23     22 

19     32 

27     34 

27     36 

11      29 

44     20 

32     36 

0     26 

52       7 

33     45 

6     35  S. 

62     28  E. 

34     28N. 

contact. 


APPENDIX. 


Eclipse  begins  at 
Sunset. 

Eclipse  ends  at  Sunrise. 

Longitude 

Latitude. 

Longitude 

Latitude. 

Longitude 

Latitude, 
i 

o       / 
63     30E. 
70    58 
77      6 
79    13 
69    17  E- 

0          / 

34    41  N. 
40       8 
63    58 
63     47 
83     59N. 

12    14W 
97     55 
98    31 
96    21 
91     15 
88    48W 

0          / 

87    50N. 
80    47 
67       3 
69    58 
43     50 
34     36N. 

o       / 

86     42  W 
81     17 
78      9 
74    67 
67    61 
63     65W 

0          / 

26     26N.I 
6     43N. 
4     148- 
12      7 
21     64 
23     378. 

These  elements  also  give  an  Annular  Eclipse  through 
England,  central  at  the  following  points  of  latitude  and 
longitude. 

Suggestion*      If  a  student  wishes  to  try  his  skill  at  projecting  local 
io  the   itu-  Eclipses,  let   him   take   the   latitude,  and   corresponding 
longitude,  from  this  table,  and  if  he  is  successful,  the  cen- 
ters of  the  sun  and  moon  will  fall  on  the  same  point  at  the 
time*  of  greatest  phase. 

The  times  here  mentioned  are  the  times  of  central 
eclipse. 


Central  line 

tomeiinfrom 
the  Atlantic 
at  56m. 

Ezito  into 
tie  North 
0eaatlh.2M. 


Greenwich  Mean  Times. 

Longitude. 

Latitude. 

Duration 
of 
The  Ring. 

d      ft  .  ,m     • 

0          / 

0         / 

sec. 

Mar.  15     0     54     0 

4  12.4  W. 

49  39.4N. 

8.1 

0     55     0 

3  40.6 

50     4.5 

8.7 

0     56     0 

3     7.7 

50  29.9 

9.3 

0     57     0 

2  33.7 

50  55.7 

9.9 

0     58     0 

1   58.7 

51   21.3 

10.5 

0     59     0 

1   22.4 

51   47.3 

11.1 

1       0     0 

0  44.7 

52   14.0 

11.7       I 

1        1     0 

0     5.8  W. 

52  40.8 

12.3 

1       *     9 

0  34.6  E. 

53     7.9 

12.9 

1       3     0 

1    16.7E. 

53  35.6N". 

13.5 

APPENDIX.  3* 

LUNAR  PERTURBATIONS.  -  A  FRAGMENT. 

Students  in  astronomy  very  naturally  infer  that  the  at- 
tractions of  Venus  and  Jupiter,  (when  thost.  planets  are 
nearest  to  the  earth,)  affect  the  longitude  of  the  moon. 

Indeed,  some  have  been  so  confident  of  this  as  to  infer       Do   1|W 
that  some  of  the  lunar  equations  were  the  results  of  such  planets   &». 
planetary  attractions.     But  we  assure  them  it  is  not  so  —  motj0nj 
none  of  the  lunar  equations  refer  to  any  other  disturbing 
body  than  the  sun. 

Yet  all  other  bodies  do  disturb  the  lunar  motion,  but  the 
amount  of  other  disturbing  forces  is  absolutely  insensible, 
as  we  shall  show  by  the  following  analysis  : 

The  lunar  perturbations  by  the  action  of  the  sun,  arise, 
as  we  have  seen  by  the  various  applications  of  the  expres-  How  to 

n  decide      the 

sion  --  ,  in  which  S  represents  the  mass  of  the  sun,  and  que«tu». 

2a3 

a  the  distances  of  the  sun  from  the  earth,  which  is  1  at  its 
mean  distance. 

But  the  greatest  effect  that  this  expression  has,  is  that 
of  changing  the  eccentricity  of  the  moon's  orbit,  which 
change  will  affect  the  moon's  longitude  to  the  amount  of 
1°  20',  or  of  80'. 

Now  in  the  place  of  S,  the  mass  of  the  sun,  in  the  ex- 

pression -  ,  let  us  take  the  mass  of  Venus,  and  let  a  rep- 

2a3 

resent  the  nearest  distance  between  Venus  and  the  earth. 
For  the  sun  the  mean  value  of  a=  1  .  But  for  Venus,  when 
that  planet  is  nearest  to  the  earth  in  mean  distance,  a=l  — 
0.7233=0.2767. 

The  mass  of  Venus  is  about  the  same  as  that  of  the  earth, 
and  when  the  earth  is  1,  the  sun  is  354936  (see  page  188). 
That  is,  if  P=l,  5=354936.  Now  let  the  effect  of 
Venus  on  the  lunar  motion  be  x,  then  we  shall  have  the 
following  proportion  : 


:    80' 


2a3  2(0.2767)3 

- 
(0.2767) 


Or         354936  :  80'    :  :  -  _  :  x. 

8 


362  APPENDIX. 

Or         4446  :   1'  :  :  1 :  x. 

(0.2767)3 

x= ,  which  reduced,  gives  a  very  §maiv 

4446(0.2767)3 

fraction  of  a  second,  for  the  value  of  a-. 

By  Jupiter.  The  mass  of  Jupiter  is  about  320  times  that  of  Venus, 
but  its  nearest  distance  is  about  15  times  the  nearest  dis- 
tance of  Venus;  therefore,  the  effect  of  Jupiter  on  the  lon- 
gitude of  the  moon  can  never  be  greater  than 

60".320 

"4446(0.^767T3(15)3 

Value*  in-  which  js  but  a  small  part  of  a  second,  and  of  course  insen- 
sible.  Therefore,  the  perturbations  produced  on  the 
moon's  motion  by  the  planets,  are  not,  and  need  not 
be  taken  into  account. 


TRANSITS    AND    OCCULTATIONS    OF    JUPITER'S    SATELLITES. 

Trantiti  of  This  appendix  is  designed  for  fragments  of  useful  astro- 
»at.  nomical  knowledge.  And  we  believe  that  the  following 
"of explanation  for  computing  the  approximate  times  for  the 
their  shad-  transits  of  Jupiter's  moons  across  the  planet,  and  the  tran- 
ce0™ tiT  8*ts  °^  their  shadows  also  across  the  face  of  the  planet,  and 
planet.  HOW  their  occultations  behind  the  planet,  will  be  highly  appre- 
oomputed.  cjate^  by  all  true  scholars. 

The  eclipses  of  these  satellites  are  computed  from  the 
tables  constructed  on  purpose  — and  they  Jill  a  large  volume. 
We  do  not  pretend  to  compute  these,  but  take  them  as  they 
are  given  in  the  Nautical  Almanac,  and  from  them  compute 
the  transits  and  occultations,  using  the  mean  motion  of  the 
satellite.  In  respect  to  the  first  satellite,  our  results  are 
Lable  to  an  error  of  about  4  minutes  —  but  usually,  they 
will  be  much  less. 

The  Immersion  andEmmersion  of  the  same  eclipse  of 
Jupiter's  first  satellite  are  never  visible  from  the  earth,  be- 
cause one  side  or  the  other  of  the  placet's  shadow  is  hid 
by  the  body  of  the  planet. 

Elements.       For  this  and  other  reasons  we  must  always  compute  the 
found  ang]e  between  the  earth  and  sun,  as  seen  from  Jupiter  »V 
the  time,  or  about  the  time  in  question.    Sufficient  elemei  fc? 


APPENDIX. 

are  given  in  the  Nautical  Almanac  to  compute  this  angle 
with  ease. 

For  example,  we  take  January  1 1th,  1 858,| 
and  find  by  the  N.  A.,  that  at  that  time, 
The  longitude  of  the  sun  is         290°  50' 
Subtract  180 


353 


Lon.  of  the  earth,  seen  from^     1 10°  50' 
Longitude  of  Jupiter     -     -          47        5 

-     -        63°  45' 


The  angle  J SE 


As  log.  JE 

Is  to  sin.  63°  45'     - 

So  is  log.  SE    - 


0.667088 
-  9.952731 
— 1.902800 

^9.945537 


Angle  J,  sin.  10°  57'       9.278443 
Log.  of  JE,  and  log,  of  SE,  are  given  in  the  N.  A. 

The  next  figure  is  on  a  large  scale,    Description 

0  of  the  figure 

representing  Jupiter  in  the  center, 
its  shadow  opposite  the  sun,  and  H, 
-4,  B,  C,  D,  F,  portions  of  the  or- 
bit of  the  first,  satellite. 

The  moon  moving  in  the  direc- 
tion from  A  to  7?,  when  it  is  at  B, 
it  is  the  time  of  the  emmersion  from 
an  eclipse. 

QC  and  QE  are  lines  drawn  di- 
rect to  the  earth,  and  QD,  21^  are 
lines  nearly  parallel  to  the  sun. 

When  the  satellite  arrives  at  C,  it     Whenth* 
is  seen  projected  on  the  face  of  the  J^^nd  *«x" 
planet.     When  it  arrives  at  D,  its  plained, 
shadow  is  cast  on  the  planet.   When 
it  arrives  at  E,  the  transit  of  the 
satellite  is  completed,  as  seen  from 
earth,  and  when  it  arrives  at  F,  the 
transit  of  the  shadow  is  completed. 

When  the  satellite  passes  on  and 


354  APPENDIX. 

arrives  at  H,  it  apparently  goes  behind  the  pUnet,  as  it  is 
said  to  be  occult,  but  it  is  not  eclipsed,  properly  speak- 
ing, until  it  arrives  at  A.  But  the  satellite  is  not  then 
visible,  and  this  immersion  into  the  shadow  is  not  noticed 
in  the  Nautical  Almanac.  The  emmersionsat  J5,  are  com- 
puted, even  to  seconds.  Our  attention  is  here  confined  to 
the  first  satellite,  but  the  principles  are  the  same  for  all. 

The  radius  of  the  planet  is  l,and  the  radius  of  the  orbit 
is  6.0485.  These  will  give  us  the  arcs  AB,  &c. 

BD  and  AF,  are  lines  to  the  sun  nearly  parallel,  and  by 
obvious  computation,  we  find  that  the  arc  AB  on  the  orbit 
=  18°  60',  DF&ud  CE  each  18°  66'. 

The  angle  CQD  is  constantly  changing,  but  at  this  tim» 
it  is  10°  57'.  Hence,  (7^=29°  53'.  £(7=180°—  29°  5& 
=  160°  7'. 

The  first  satellite  passes  over  360°  in  42h.  14m.  28s. 

Therefore  it  passes  orer  the  following  arcs  on  its  ^rbit 
in  the  times  set  opposite  to  them. 

The  times  are  given  in  hours,  minutes,  and  hundreds  of 

a  minute. 

Arcs.      Time.  Arcs.        Time. 

h  140°=  16  31.20 

rc.poncling.  %#__.    g  33  7o_      4953  160°=  17  42 

300°==  35  24 


10'=   1.18 

6°=    42.48 

20'=  2.36 

7°=    49.56 

30'=  3.54 

8°=     5662 

40'=  4.72 

9°=1     3.72 

60'=  5.90 

10°=1   iO.8 

1°=  7.08 

11°=1  17.88 

2°=14.16 

3°=21.24 

4°=28.32 

19°=2  14.52 

5°=35.40 

20°=2  21.60 

By  the  Nautical  Almanac  we  find  that  the  first  satellite 
passed  B,  and  reappeared  from  an  eclipse,  mean  Greenwich 

time, 

h.    m.     s 

Jan.  10,  0  36  10.4 

From  B  round  to  C  150°  7'=         17  43  nearly. 


APPENDIX. 

d.         h.      m. 

Transit  of  eat.,  Immersion,  10  18  19 
Result  in  Naut.  Almanac  is  10  18  24 
From  (7  to  .0=10°  57'=  1  17 


Transit  of  shadow,  Immers.   10     19  41  same  in  N.  A. 
From  J)toE7°  59'=  56 


Transit  of  satellite,  Em.         10     20  37     36m.  in  N.  A. 
From  E  to  F  10°  57'  1   17 


Transit  of  shadow,  Em.  10     21   54     53m.  in  N.  A. 

From  Fio  H  150°  T  17  43 


Occultation  of  satellite  11     1537     same  in  N.  A. 

The  satellite  will  not  be  seen  again  until  it  passes  B,  or 
passes  over  an  arc  which  we  have  estimated  at  29°  53',  or  computation 
in  time  3h.  31m.     This  added  to  the  last  result  will  give 
lid.  19h.  8m.  for  the  reappearance  of  the  satellite.     By  the 
tables  it  is  lid.  19h.  5m.  14s. 

We  can  now  go  over  the  revolution  again,  after  finding 
a  new  value  for  the  angle  CQD,  and  thus  we  might  pro- 
ceed over  any  number  of  revolutions,  and  with  any  one  of 
the  satellites. 


Example 


ELEMENTS   OF   THE   ECLIPSES   OF   THE   SUN 

1846.                          April  25. 

b.   m.    t. 

October  19. 

h.    m.     •. 

Greenwich  M.  T.  of  <J  in  R.  A., 

4  55  54  -5 

195012.2 

O  and  (§'s  Right  Ascension, 

2  11    8  -31 

13  38  31  -54 

0           /          n 

O           1         II 

O's  declination,                    N. 

13  25  19  -8 

S.  10  23  43  -0 

O's  declination,                     N. 

13  13  21  -2 

S.1015    3-9 

9  's  hourly  motion  in  R.  A., 

33  55  -1 

30  42  -2 

O's  hourly  motion  in  R.  A., 

221-3 

221-5 

(§  's  hourly  motion  in  dec.    N. 

823-6 

S.        8  37  -0 

O's  hourly  motion  in  dec.    N. 

048-8 

S.        054-1 

(§'s  equatorial  hor.  parallax, 

57  53  -8 

5533-4 

O's  equatorial  hor.  parallax, 

8-5 

8-6 

^  's  true  semidiameter, 

15  46  -5 

15    8-4 

O's  true  semidiameter, 

15  54  -5 

16    5-6 

356  APPENDIX. 

THE    APRIL    ECLIPSE. 

Begjns  on  the  earth  generally  April  25  d.  2  h.  2  m.  4  s.,  mean 

time  at  Greenwich,  in  longitude  119°  40'  W.  of  Greenwich, 

and  latitude  6°  15'  S. 
Central   Eclipse  begins  generally  April   25  d.  3  h.  3  m.  3  B. 

in  longitude  135°  51'  W.  of  Greenwich,  andlat.  2°  11'  S. 
Central  eclipse  at  noon,  April  25  d.  4  h.  55  m.  9  a. 

in  longitude  74°  31'  W.  of  Greenwich,  and  lat.  25°  21'  N. 
Central  eclipse  ends  generally  April  25  d.  6  h.  37  m.  6s, 

in  longitude  3°  43'  W.  of  Greenwich,  and  lat.  24°  56'  N. 
Ends  on  the  earth  generally  April  25  d.  7  h.  38  m.  5s, 

in  longitude  20°    4'  W.  of  Greenwich,  and  lat.  20°  52'  N. 

THE   OCTOBER   ECLIPSE. 

Begins  on  the  earth  generally  October  19  d.  16h.  46m.  7s. 
mean  time  at  Greenwich,  in  longitude  16°  21'  E.  of  Green- 
wich, and  latitude  9°  50'  N. 

Central  eclipse  begins  generally  October  19  d.  17  h.  52m.  0  s. 
in  longitude  0°  32'  W.  of  Greenwich,  and  lat.  6°  44' N. 

Central  eclipse  at  noon,  October  19  d.  19  h.  50  m.  2  s 

in  longitude  58°  41'  E.  of  Greenwich,  and  lat.  19°  22'  S. 

Central  Eclipse  ends  generally  October  19  d.  21  h.  38  m.  9  s. 
in  longitude  126°  5'  E.  of  Greenwich,  and*  lat.  23°  51'  S. 

Ends  on  the  earth  generally  October  19  d.  22  h.  44  m.  1  s. 
in  longitude  109°  6'  E.  of  Greenwich,  and  lat.  20°  47'  S. 

The  following  is  a  catalogue  of  the  solar  eclipses  that  will 
be  visible  in  New  England  and  New  York,  between  the  years 
1850  and  1900;  the  dates  are  given  in  civil,  not  astronomi- 
cal, time. 

statistics  1851^  juiy  28th.    Digits  eclipsed,  3f ,  on  sun's  northern  limb 
from  Tsso  to  1854,  May  26th.     As  computed  in  the  work. 
MOO.  1858,  March  15th.     Sun  rises  eclipsed.     Greatest  obscura- 

tion, 5i  digits  on  sun's  southern  limb. 

1859,  July  29th.    Digits  eclipsed,  2i,  on  sun's  northern  limb. 

1860,  July  18th.     Digits  eclipsed,  6,  on  sun's  northern  limb. 

1861,  December  31st.     Sun  rises  eclipsed.     Digits  eclipsed 

at  greatest  obscuration,  4i,  on  sun's  southern  limb. 


APPENDIX.  357 

1865,  October  19th.     Digits  eclipsed,  8j,  on  sun's  southern 

limb. 

1866,  October  8th.    1  digit  eclipsed.     South  of  New  York 

no  eclipse. 
1869,  August  7th.    Digits  eclipsed,  10,   on  sun's  southern 

limb.     This  eclipse  will  be  total  in  North  Carolina. 
1873,   May  25th.     Sun   and   moon   in   contact   at  sunrise, 

Boston. 

1875,  September  29th.     Sun   rises   eclipsed.     This   eclipse 

will  be  annular  in  Boston,  Maine,  New  Hampshire, 
and  Vermont. 

1876,  March  25th.     Digits    eclipsed,   3£,  on  sun's  northern 

limb. 
1878,  July  29th.     Digits   eclipsed,    7£,   on    sun's   southern 

limb.     This  is  the  fourth  return  of  the  total  eclipse 

of  1806. 
1880,  December  31st.     Sun  rises  eclipsed.     Digits   eclipsed 

at  greatest  obscuration,  5^,  on  sun's  northern  limb. 

1885,  March  16th.     Digits  eclipsed,   6^,   on  sun's  northern 

limb. 

1886,  August  28th.     North   of  Massachusetts  no  eclipse, 

south,  sun  eclipsed.  statistic! 

1892,  October  20th.    Digits  eclipsed,  8,  on  sun's  northern  j£ 

limb.  looo. 

1897,  July  29th.    Digits  eclipsed,   4i,  on   sun's  southern 

limb. 
1900,  May  28th.    Digits  eclipsed,   11,   on  sun's  southern 

limb.    The  sun  will  be  totally  eclipsed  in  the  State 

of  Virginia. 


TO 


CONTENTS    TO   APPENDIX. 


A  more  general  exposition  of  the  solar  eclipse  of  May,  1854, 
given  in  respect  to  its  entire  appearance  over  the  earth. 
L         The  elements  being  taken  from  the  English  Nautical  Al- 
manac, in  terms  of  Right  Ascension  and  Declination,. . . .         310 
The  necessary  preparations  for  the  computation  in  all  its 

particulars, 311 

Computations  of  the  latitudes  of  the  several  points  of  begin- 
ning, ending,  &c 316 

Problems  of  Longitudes  of  the  same  points, 320 

Lon.  and  Lat.  of  any  assumed  point, 327 

Minute  calculations  respecting  local  eclipses, 330 

Rising,  passing  meridian,  and  setting  of  the  stars,  planets, 

and  moon,  —  how  computed, 336 

Arguments  for  the  small  equations  of  the  moon  in  LoLgitude 

and  Latitude,  noted, 345 

Elements  of  the  solar  eclipse  in  March,  1858. — Given  as  an 

example,  343 

Lunar  Perturbations. — A  Fragment 351 

Transits  and  Occupations  of  Jupiter's  Satellites, 352 

Catalogue  of  eclipses  which  will  take  place  between  the  yerrs 
1850andl900 355 

Table  of  the  Asteroids,  on  page  55  of  Tables. 

368 


TABLES.  i 

EXTRACTS   FROM   THE   NAUTICAL   ALMANAC   FOR    JANLARY,    1846. 


THE  SUN'S 

Logar. 

(  f  L 

I 

Apparent 

01  tne 
ladius 

THE  MOON'S 

0fc 

Vector 

j: 
o 

5* 

Longitude. 

Latitude, 

of  the 
Earth. 

jongitude. 

Latitude. 

Semi- 
diam. 

Hor. 
Paral. 

a 

Noon. 

Noon. 

Noon. 

Noon. 

Noon. 

Noon. 

Noon. 

1 

O        1        II 

280  46  15.3 

it 

N.0.49 

9.99266 

0                  // 

30  42  13.9 

0       /         II 

N.4  54    8.5 

i    a 
16  21.6 

/     a 
60    2.3 

"I 

281  47  26.1 

0.45 

9.992G6 

45    7  12.0 

4  24    8.7 

16    8.3 

59  13.5 

3 

282  48  36.5 

0.37 

9.99267 

59    4  55.4 

3  39    5.9 

15  53.9 

58  20.5 

4 

283  49  46.5 

0.27 

J.99267 

12  35  34.7 

2  43    1.9 

15  39.S 

57  28.7 

m 

284  50  56.1 

0.1G 

9.H9-268 

25  41  31.5 

1  39  55.7 

15  26.7 

56  40.8 

6 

285  52    5.3 

N.0.03 

9.99268 

38  26  25.0 

N.O  33  28.3 

15  15.2 

55  58.7 

7 

286  53  13.9 

S.0.11 

9.9927(1 

50  54  23.2 

S.O  33    36 

15    5.6 

55  23.3 

8 

287  54  22.0 

0.25 

9.99271 

63    9  30.1 

1  36  46.8 

14  57.6 

54  54.1 

y 

288  55  29.7 

0.38 

9.99272 

75  15  21.8 

2  35    8.6 

14  51.5 

54  31.6 

10 

•289  56  36.8 

0.49 

9.99274 

87  14  56.3 

3  25  55.4 

14  46.9 

54  14.6 

1 

290  5V  43.4 

0.58 

9.99277 

99  10  31.3 

4    7  13.7 

14  43.8 

54    3.3 

• 

291  58  49-5 

0.65 

9.99279 

111     3  50.8 

4  37  30.7 

14  42.1 

53  57.0 

3 

292  59  55.3 

0.70 

9.99282 

122  56  17.6 

4  55  38.9 

14  41.7 

53  55.7 

294    1     0.5 

0.71 

9.99285 

134  49    7.9 

5    0  5G  4 

14  42.8 

53  59.8 

&)5    2    5.4 

0.69 

9.9926e 

146  43  48.4 

4  53    7.6 

14  45.5 

54    9.7 

j 

296    3    9.9 

0.64 

9.99292 

158  42  11.3 

4  32  23.1 

14  50.( 

5426.0 

297    4  14.0 

0.57 

9.99295 

170  46  44.8 

3  59  17.1 

14  56.3 

54  49.0 

Ib 

298    5  17.8 

0.47 

9.99299 

183    0  38.7 

3  14  47.1 

15    4.6 

55  19.7 

19 

299    6  21.2 

0.35 

9.99304 

195  27  41.8 

2  20  14.2J13  15.2 

55  58.4 

20 

300    7  24.2 

0.23 

9.99308 

208  12  10.4 

1  17  27.815  27.7 

56  44.4 

2 

301     8  26.7 

S.  0.09 

9.99313 

221  18  27  5 

S.O    8  53;1  15  42.057  37.0 

1 

22 

302    9  2F.9 

N.0.04 

9.99318 

234  50  26.7|N.l     2  20.5J15  57.3|58  32.9 

2d 

303  10  30.4 

0.15 

9.99323 

248  50  42.5 

2  12  11.7116  12.5J59  28.8 

24 

304  11  31  .3j       0.25 

'K99328 

263  19  30.4 

3  15  50.9 

16  26.2 

60  1».0 

25 

305  12  31.5 

0.33 

9.99334 

278  13  48.8 

4    8    2.8 

16  36.8 

60  57.9 

26 

306  13  30.9 

0.38 

9.99339 

293  26  49.2 

4  43  49.4 

16  42.9 

61  20.2 

27 

307  14  29.3 

0.40 

9.99345 

308  48  22.8 

4  59  32.4 

16  43.5 

61  22.6 

28 

308  15  26.8 

0.40 

9.9935 

324    6  34.0 

4  53  45.4 

16  38.7 

61    4.9 

29 

309  16  23.3 

0.37 

9.9935 

339    9  55.3 

4  27  32.9 

16  28.8 

60  29.1 

3 

310  17  1&.5 

0.30 

9.9936 

353  49  32.0 

3  44    8.2 

16  15.6 

59  40.2 

3 

311  18  12.6 

0.21 

9.9936 

8    0  13.1 

2  47  5o.7 

16    O.S 

58  43.7 

32 

312  19    5.3 

N.0.10 

9.99375 

21  40  34.3 

N.I  43  50.C 

15  44.2 

57  45.1 

TABLES. 


TABLE    I. 

MEAN   ASTRONOMICAL   REFRACTIONS. 
Barometer  30  in.     Thermometer,  Fah.  50°. 


AR.A., 

Rofr. 

Ap.  Alt. 

Ref'r. 

Ap.  Alt. 

Refr. 

Alt. 

Refr. 

0°  0' 

33'  51" 

4°  0' 

11'  52" 

12°  0' 

4'  28.1" 

42° 

1  4.6' 

5 

32  53 

10 

11  30 

10 

4  24.4 

43 

1  2.4 

10 

31  58 

20 

11  10 

20 

4  20.8 

44 

1  0.3 

15 

31  5 

30 

10  50 

30 

4  17.3 

45 

0  58.1 

20 

30  13 

40 

10  32 

40 

4  13.9 

46 

56.1 

25 

29  24 

50 

10  15 

50 

4  10.7 

47 

54.2 

30 

28  37 

5  0 

9  58 

13  0 

4  7.5 

43 

52.3 

35 

27  51 

10 

9  42 

10 

4  4.4 

49 

50.5 

40 

27  6 

20 

Q  27 

20 

4  14 

50 

48.8 

45 

26  24 

30 

9  11 

SO 

3  58.4 

51 

47.1 

50 

25  43 

40 

8  58 

40 

3  55.5 

52 

45.4 

55 

25  3 

50 

8  45 

50 

3  52.6 

53 

43.8 

1  0 

24  25 

6  0 

8  32 

14  0 

3  49.9 

54 

42.2 

5 

23  48 

10 

8  20 

10 

3  47.1 

55 

40.8 

10 

23  13 

20 

8  9 

20 

3  44.4 

56 

39.3 

15 

22  40 

30 

7  58 

30 

3  41.8 

57 

37.8 

20 

22  8 

40 

7  47 

40 

3  39.2 

58 

36.4 

25 

21  37 

50 

7  37 

50 

3  36.7 

59 

35.0 

30 

21  7 

7  0 

7  27 

15  0 

3  34.3 

60 

33.6 

35 

20  38 

10 

7  17 

15  30 

3  27.3 

61 

32.3 

40 

20  10 

20 

7  8 

16  0 

3  20.6 

62 

31.0 

45 

19  43 

30 

6  59 

16  30 

3  14.4 

63 

29.7 

50 

19  17 

40 

6  51 

17  0 

3  8.5 

64 

28.4 

55 

18  52 

50 

6  43 

17  30 

3  2.9 

65 

27.2 

2  0 

18  29 

8  0 

6  35 

18  0 

2  57.6 

66 

25.9 

5 

18  5 

10 

6  28 

19 

2  47.7 

67 

24.7 

10 

17  43 

20 

6  21 

20 

2  38.7 

68 

23.5 

15 

17  21 

30 

6  14 

21 

2  30.5 

69 

22.4 

20 

17  0 

40 

6  7 

22 

2  23.2 

70 

21.2 

25 

16  40 

50 

6  0 

23 

2  16.5 

71 

19.9 

30 

16  21 

9  0 

5  54 

24 

2  10.1 

72 

18.8 

35 

16  2 

10 

5  47 

25 

2  4.2 

73 

17.7 

40 

15  43 

20 

5  41 

26 

1  58.8 

74 

16.6 

45 

15  25 

30 

5  36 

27 

53.8 

75 

15.5 

50 

15  8 

40 

5  30 

28 

49.1 

76 

14.4 

55 

14  51 

50 

5  25 

29 

44.7 

77 

13.4 

3  0 

14  35 

10  0 

5  20 

30 

40.5 

78 

12.3 

5 

14  19 

10 

5  15 

31 

36.6 

79 

11.2 

10 

14  4 

20 

5  10 

32 

33.0 

80 

10.2 

15 

13  50 

30 

5  5 

33 

29.5 

81 

9.2 

20 

13  35 

40 

5  0 

34 

26.1 

82 

8.2 

25 

13  21 

50 

4  56 

35 

23.0 

83 

7.1 

30 

13  7 

11  0 

4  51 

36 

20.0 

84 

6.1 

35 

12  53 

10 

4  47 

37 

17.1 

65 

5.1 

40 

12  41 

20 

4  43 

38 

1  14.4 

86 

4.1 

45 

12  28 

30 

4  39 

39 

1  11.8 

87 

3.1 

50 

12  16 

40 

4  35 

40 

1  9.3 

88 

2.0 

55 

12  3 

50 

4  31 

41 

1  6.9 

89 

1.0 

TABLE   C. 

CORRECTION  OP  MEAN  REFRACTION. 
Hight  of  the  Thermometer. 


App. 

240 

280 

320 

36° 

400 

440 

520 

560 

600 

640 

680 

72° 

760 

800 

Alt. 

0  ' 

0.10 

2.18 

1.55 

1.33 

1.11 

51 

31 

10 

29 

48 

1.07 

1.25 

1.43 

2.01 

2.19 

0.00|2.12|1.49 

1.28 

1.08 

48 

29 

9 

27 

45 

1.04 

1.21 

1.38 

1.54 

2.12 

0.20 

2.05 

1.44 

1.24 

1.04 

46 

28 

9 

26 

44 

1.01 

1.17 

1.33 

1.49 

2.05 

0.30 

1.59 

.39 

1.20 

1.01 

44 

26 

8 

25 

41 

58 

1.13 

1.28 

1.43 

1.59 

0.40 

1.53 

.34 

1.16 

58 

42 

25 

8 

24 

39 

55 

1.10 

1.24 

1.38 

1.53 

0.50 

1.48 

.29 

1.12 

55 

40 

24 

8 

23 

37 

52 

1.06 

1.20 

1.34 

1.48 

1.00 

1.43 

.25 

1.09 

53 

38 

23 

7 

21 

36 

50 

1.03 

1.17 

1.30 

1.43 

1.  10 

1.38 

.21 

1.06 

50 

36 

22 

7 

20 

34 

48 

1.00 

1.13 

1.26 

1.38 

1.20 

1.33 

.17 

1.03 

48 

34 

21 

6 

19 

32 

45 

57 

1.09 

1.21 

1.33 

1.30 

1.29 

.14 

1.00 

46 

32 

20 

6 

18 

31 

43 

54 

1.06 

1.18 

1.29 

1.40 

1.25 

.11 

57 

44 

31 

18 

6 

18 

30 

41 

52 

1.04 

1.15 

1.25 

1.50 

1.21 

.08 

55 

42 

30 

17 

6 

17 

28 

39 

50 

1.01 

1.11 

1.21 

2.00 

1.18 

1.05 

53 

39 

29 

17 

5 

16 

27 

37 

48 

58 

1.08 

1.18 

2.20 

1.11 

1.00 

48 

37 

26 

16 

5 

15 

25 

35 

44 

54 

1.03 

1.11 

2.40 

1.06 

55 

44 

34 

24 

14 

5 

14 

23 

32 

41 

50 

58 

1.06 

3.00 

1.01 

51 

41 

32 

22 

13 

4 

13 

21 

30 

38 

46 

54 

1.01 

3.20 

57 

47 

38 

29 

21 

13 

4 

12 

20 

28 

35 

43 

50 

57 

3.40 

53 

44 

36 

28 

20 

12 

4 

11 

18 

26 

33 

40 

47 

53 

4.00 

49 

41 

33 

26 

18 

11 

4 

10 

17 

24 

31 

37 

44 

50 

4.30 

45 

38 

31 

24 

17 

10 

3 

9 

16 

22 

28 

34 

40 

45 

5.00 

41 

35 

28 

22 

16 

9 

3 

9 

14 

20 

26 

31 

36 

40 

5.30 

38 

32 

26 

20 

14 

9 

3 

8 

13 

19 

24 

29 

34 

38 

6.00 

35 

30 

24 

19 

13 

8 

2 

7 

12 

17 

22 

26 

31 

35 

6.30 

33 

28 

22 

17 

12 

7 

2 

7 

11 

15 

20 

24 

29 

33 

7.00 

31 

26 

21 

16 

12 

7 

2 

6 

10 

14 

19 

23 

27 

31 

8 

27 

23 

19 

15 

10 

6 

2 

5 

9 

13 

16 

20 

24 

27 

9 

24 

20 

16 

13 

9 

5 

2 

5 

8 

11 

14 

18 

21 

24 

10 

22 

18 

15 

12 

8 

5 

4 

7 

10 

13 

16 

19 

22 

11 

20 

17 

14 

11 

8 

5 

4 

7 

9 

12 

15 

18 

20 

12 

18 

15 

13 

10 

7 

4 

4 

6 

9 

11 

13 

16 

18 

13 

17 

14 

12 

9 

7 

4 

3 

6 

8 

10 

12 

15 

17 

14 

16 

13 

11 

8 

6 

4 

3 

5 

7 

9 

11 

14 

16 

15 

15 

12 

10 

8 

6 

3 

3 

5 

7 

9 

11 

13 

15 

16 

14 

12 

9 

7 

5 

3 

3 

5 

6 

8 

10 

12 

14 

17 

13 

11 

9 

7 

5 

3 

3 

4 

6 

8 

9 

11 

13 

18 

12 

10 

8 

6 

5 

3 

2 

4 

6 

7 

9 

10 

12 

19 

11 

9 

8 

6 

4 

3 

2 

4 

5 

7 

8 

10 

11 

20 

11 

9 

7 

6 

'4 

2 

2 

4 

5 

6 

8 

9 

11 

21 

10 

9 

7 

5 

4 

2 

2 

3 

5 

6 

7 

9 

10 

22 

10 

8 

7 

5 

4 

2 

2 

3 

5 

6 

7 

8 

10 

23 

9 

8 

6 

5 

4 

2 

2 

3 

4 

6 

7 

8 

9 

24 

9 

7 

6 

5 

3 

2 

2 

3 

4 

5 

6 

8 

9 

25 

8 

7 

6 

5 

3 

2 

1 

2 

3 

4 

5 

6 

7 

8 

26 

8 

7 

6 

4 

3 

2 

1 

2 

3 

4 

5 

6 

7 

8 

27 

8 

6 

5 

4 

3 

2 

1 

2 

3 

4 

5 

6 

7 

8 

28 

7 

6 

5 

4 

3 

2 

0 

1 

2 

3 

5 

5 

6 

7 

30 

7 

6 

5 

4 

3 

2 

0 

I 

2 

3 

4 

5 

6 

7 

28.26 

28^56 

28.85 

2915 

29.75 

30.05 

30.35 

30~fei 

30.93 

Hight  of  the  Barometer. 

TABLES. 
TABLE    II. 

MEAN  PLACES  FOR  100  PRINCIPAL   FIXED  STARS,  FOR  JAN.  I,  1S4«. 


Star's  Nam«. 

* 

S 

Right  Ascen. 

Annual  Var 

Declination. 

Ann.  Var.  ] 

+20C.055  . 
20.050 
19.997 
19.862 

+19.810 
19.279 
18.952 
18.461 

4-17.432 
15.621 
14.532 
13.329 

4-11.620 
10.711 
7.097 
4.737 

4-  4.583 
3.776 
3.123 
2.968 

4-  2.754 
2.262 
4-  1.149 
—  1.196 

—  1.796 
2.337 

4.484* 
4.562 

—  6.110 
7.253 

8.758« 
8.152 

—10.104 
12.800 
13.464 
14961 

—15.366 
16.108* 
16.283 
—17.377 

«  ANDROMED./E    

1 
9  3 

0  0  26.257 
0  5  18.691 
0  17  34.168 
0  31  48.294 

0  35  51.339 
1  3  52.226 
1  16  19.692 
1  31  58.291 

1  58  30.193 
2  35  19.633 
2  54  14.072 
3  13  21.403 

3  33  20.382 
3  50  50.760 
4  27  5  404 
5  5  19.317 

5  7  8.383 
5  16  33.662 
5  24  8.428 
5  25  56.406 

5  28  24.062 
5  34  4531 
5  46  50.189 
6  13  38.621 

6  20  32.145 
6  26  30.287 
6  38  21.883 
6  52  34.440 

7  10  55.298 
7  24  46.065 
7  31  14237 
7  35  53.153 

8  0  59.232 
8  38  37  154 
8  48  38.088 
9  12  58.192 

9  20  1.170 
9  22  31.453 
9  37  6.098 
0  0  10.062 

4-  3.0720 
3.0784 
1      3.3054* 
3.3418 

4-  2.9995 
17.1346* 
3.0015 
22339 

4-  3.3475 
3.1085 
3.1266 
4.2324 

-4-  3.5473 

2.7898 
3.4274 
4.4082 

4-  2.8787 
3.7827 
3.0609 
2.6425 

+  3  0404 
2.1691 
3.2433 
3.6257 

-|-  1.3279 
30.7946 
2.6459* 
2.3558 

+  3.5918 
3.8561 
3.1445* 
3.6829* 

+  2.5596 
3.1966 
4.1261* 
1.6100 

+  2.9499 
4.0504* 
3.4258 
-t-  3.22  11 

deg      mm.    sec. 

N.28  14  25.40 
N.14  19  37.8(1 
S.78    7  24.40 
N.55  41  31.08 

S.18  49  59.01 
N.88  29  17.88 
S.   8  58  45  93 
S.58     1  14.34 

N.22  43  53.86 
N.  2  35     1.17 
N.  3  28  5570 
N.49  18  28.20 

y  PEGASI  (Algenib),..  .  . 

0  Hydri,  

3 
3 

2.3 
2.3 
~3 

1 

3 
3 
2.3 
2.3 

3 

o  «i 

a.  CASSIOPE,*,  

£  Ceti,  

«  URS.  Mix.  (Polaris),. 
6>  Ceti  

*  Eridam  (Achernar),. 

«t   AlUETIS 

y  Ceti,  

a  Cirrr  

N.23  37  27.73 
S.  13  57     1.50 
N.16  11  41.39 
N.45  50    6.56 

S.   8  23    3.33 
N.28  28  17.49 
S.   0  25    4.86 
S.  17  56  12.77 

S.    1  18  17  53 

•  i  TSridaui 

a  TAURI,  (Aldebaran),.. 
a.  AUUJU/E,  (Capella),..  . 

ft  ORIOXIS,  (#i?eJ)  
/3  TAURI,  

1 
1 

1 
2 
2 
3.4 

2  .'- 
2 

1 
3 

1 
6 
1 

3.4 
3 

1  .2 
2 

3.4 
4 
3.4 
2 

2 
3 
3 

1 

S.34    9  .S6.95 
N.  7  22  22.32 
N.22  35  13.16 

S.52  36  49.17 
N.87  15  31.20 
S.  16  30  32.83 
S.28  45  59.38 

N.22  15  37.47 
N.32  13  12.93 
N.  5  36  5495 
N.28  23  3406 

S.23  51  50.94 
N.  6  58  48.51 
N.48  38  32  35 
S.58  37  49.78 

S.    7  59  39.05 
N.52  22  31.09 
N.24  28  49.46 
N.12  43    2.96 

*  Argus,  (Ca.nopus),.  .  . 
51  (Hev.)  Cephei 

«  CANIS  MAJ.,  (Sirius), 
t  Canis  Majoris,  

«f  Gerninorurn,  

o2  GEMINOR.  (Castor),... 
A  CAN.  Mix.,  (Procyon), 
/8  GEMINOR,  (Pollux),.. 

15  Argus,  

i  Hydrse  

*  Ursje  Majoris,  

*  HYDR^E      .  . 

fl  Ursae  Majoris,  

•  Leonis,  

*  LKONIS,  (Rcgulus},.  .  . 

TABLE   II. 


Star's  Name. 

ti 

i 
?. 

Right  Ascen. 

Annual  Var. 

Declination. 

Ann.  Var. 

2 

f\ 

3 
.4 

i 

2 
5 
1 

I 

3 
1 

1 
1 

3 
3 
3 

2  '• 

2 

2  .i 
4 
2 

3 
J 
3 
2 

4 
3. 
6 
2 

2 
2 
3. 
3 

1 
3 
3 
3. 

3 

3.< 
3 

0  39     6.223 
0  54  10.737 
1     5  54  583 
1  11  38.718 

1  41  12.066 
1  45  42.219 
12    9  26.b9.s 
12  18    4916 

12  26  18465 
12  48  49007 
13  17     52ltt 
13  41  27.894 

13  47  21.140 
13  53     0.8i  '0 
14    8  38.366 
14  29  11.925 

14  38  15706 
14  42  22.132 
14  51   13.199 
15    8  43.595 

15  28  1008: 
15  36  41.07" 
15  49  41.194 
15  56  29  397 

-1-  2.3051 
3.8001 
3.1928 
30010 

+  3.0654* 
3.1874 
3.3409 
3.2710 

-i-  3.1342 
2.8403 
3.1512 
2.3525* 

H-  2.86H6 
4.1508 
2.7336* 
40165* 

+  2.6229 
4-  3.3J02 
—  0.2692 
H-  3.2226 

+  2.5279 
4-  2.9.S91 
—  2.3520 
+  3.4742 

+  3  1382 
3.6638 
0.7960 
+  6.2587 

—  6.5328* 
+  2.7320 
106.8627 
1.3513 

-h  2.7727 
1.S900 
+  3.5861 
—19.2683 

+  2.0118 
22124 
2.7566 
-f-  3.U086 

+  2.8511 
2.9254 
2.944H 
3.3315 

deg      min.  see. 

.  58  52  34.26 
N.62  34  51.81 
N.21  21  5986 
S.13  56  46.85 

NM5  25  58.12 

N  54  33    3.18 
b.78  27  26.15 
b.62  14  39.74 

S.  22  32  39.93 
\.39     9    418 
S.  I'l  21  20.80 
N.50    5     1.45 

N.19  10  21.03 
S.59  37  33.93 
N.19  59  12.H7 
S  GO  11  37.00 

N.27  43  35.23 
S.  15  23  53.52 

N.74  47    5.58 
S.   8  48  38.53 

N.27  14  11.07 
N.  6  54  49  88 
N.78  15  55.43 
S.19  22  44.18 

S.   3  17  35.67 
S.  26    5    4  58 
N.61  51  50.58 
S.  68  44    4.75 

N.82  16  52.30 
N.14  U  12.67 
S  t9  16  10.25 
N.52  25    3.28 

N.12  40  37  11 
N.5i  30  33.50 
S.  «1     5  36  14 
N.86  35  42.5e 

N.38  38  35.33 
N.33  11  14.80 
N.I  3  38  20.49 
N.  2  48  43.64 

N.10  14  31.50 
N.  8  27  54.32 
N.  6    1  33.90 
S.  13     1     4.19 

—18.33 
1924 
19.50 
1961 

—1999 

2:».02 

20.04 
19.99 

—19.92 
19.60 
18.94 
18.12 

—17.89 
17.67 
18.94* 
15.12« 

—15.46 
1523 
14.71 
13.63 

—12.33 
11.74 
10.80 
10.29 

—  9.55 

8.48 
8.o2 
7.48 

—  5.03 
4.54 
3.14 

2.88 

—  2.81 
—  0  61 
4-040 
4-  1.91 

4-  2.77 
386 
5.05 
4-  6.67 

4-8.39 

8.74 
8.55* 
10.74 

A  URS./K  MAJORIS       .  .  . 

<f  Hydrse  et  Crateris,.  . 
ft  LFONIS,  

j   UR.S^E  MAJORIS,  
$  Charnaeleontis,  
*'  Crucis    .         

&  Corvi    

12  Canum  Venaticorum, 
a.  VIRGINIS,  (Spica,)..  .  . 
H  URS^E  MAJORIS 

/8  Ct'iitauri,  

a.  Boons,  (Arciurus,).  . 
a*  Centauri    

@  UKS^E  MINORIS,  
@  Librae  

a.  CORONA  BOREALIS,..  . 

£  Ur^se  Miuoris,  
fi\  Scorpii,  

16    6  16830 

*  SCORPII,  (Antares,).. 
M  Dracouis,  

16  19  58.461 
16  21  55.119 
16  32  25.090 

17     1  55988 
17     7  37.617 
17  22  55004 
17  26  57.473 

17  27  47219 
17  53     1  955 
18    4  33.276 
18  22    0.703 

18  31  43.3^-6 
18  44  2;*.69C 
18  58  19.965 
19  17  43.889 

19  38  56.278 
19  43  16128 
19  47  44.86fc 
20    9  30.316 

«t  Trianguli  Australia, 

i  Ursse  Minoris,  
a  HERCULIS 

@  DRACONIS,  
«t  OPHIUCHI  .  .  .  . 

y  DRACONIS  •  •  • 

/*'  Sagittarii,  
i  URS^E  MINORIS,  

«  LYRJJ:,  (Vega,}  
$  LYR/E   

>  AQUIL^E,  

«  AQUIL.E,  (Altair,)..  . 

*2  CAPBICORNI,  

TABLES. 


Star's  Name. 

si 
S 

Right  Ascen. 

Annual  Var. 

Declination. 

Ann   Var. 

2 
5 

1 
5.6 

3 
3 
3 
3 

2.3 
3 
2 
3 

1 
2 
4.5 
3 

20  13  25.814 
20  16  31.309 
20  36  11.005 
20  59  59  947 

21     6  23.073 
21  14  53.940 
21  23  26.875 
21  26  39.120 

21  36  37.346 
21  57  52.326 
21  58  29.837 
22  33  46.976 

22  49    7.531 
22  57     5.584 
23  32     1.736 
23  33    4.581 

+  4.8046 
—52.1273 
+  2.0418 
2.6908* 

-h  2.5486 
1.4163 
3.1628 
O.b059 

+  2.9441 
3.0831 
3.8134 
2.9837 

+  3.3095 
2.9776 
3.0569 
+  2.4042 

S.57  13  19.50 
N.88  50  53.54 
N.44  43  57.43 
N.37  59  42.08 

N.29  35  53.03 
N.61  56  4.55 
S.  6  14  44.46 
N.69  53  7.21 

N.  9  10  17.35 
S.  1  3  56.72 
S.  47  42  12.42 
N.10  1  44  67 

S.30  26  12.28 
N.14  22  40.12 
N.  4  47  30.74 
N.76  46  22.01 

+  H.03 
11.22 
12.64 

17.48» 

+  14.57 
15.07 
15.56 
15.73 

-I-  16.26 
17.28 
17.30 
18.65 

4-  19.11 
19.31 
19.36* 
4-  19.92 

y  Ursae  Minoris,  

61>  CYGNI,  

$  CEPHEI,  

OL   AQUARIA  

a  Gruis  

*  Pis.  &.vs.(Fomalha.ut), 
a.  PEGASI  (Markab'),  

y  Cephei,  

Those  Annual  Variations  which  includes  proper  motion  are  distinguished 
by  an  Asterisk. 


SUN'S  RIGHT  ASCENSION  FOR  1846. 


By 

of 
Mo. 

January. 

February. 

March. 

April. 

May. 

June. 

1 

5 
10 
15 
20 
25 
30 

h.  min.  sec. 

18  46  52 
19  4  30 
19  26  21 
19  47  57 
20  9  17 
20  30  19 
20  51  0 

h.  min.  sec. 

20  59  11 
21  15  22 
21  35  18 
21  54  54 
22  14  12 
22  33  14 

h.  min.  sec. 

22  48  17 
23  3  12 
23  21  40 
33  40  0 
23  58  14 
0  16  25 
0  3*36 

h.  min.  sec. 

0  41  52 
0  56  26 
1  14  43 
1  33  6 
1  51  38 
2  10  22 
2  29  17 

b.  min.  sec. 

2  23  6 
2  48  25 
3  7  47 
3  27  24 
3  47  15 
4  7  20 
4  27  8 

h.  min.  sec. 

4  35  48 
4  52  12 
5  12  50 
5  33  34 
5  54  22 
6  15  10 
6  35  55 

"3 

Mo. 

July. 

August. 

September. 

October. 

November. 

December. 

1 

5 
10 
15 
20 
25 
3C 

6  40  4 
6  56  34 
7  17  5 
7  37  25 
7  57  33 
8  17  28 
8  37  7 

h.  min.  sec. 

8  44  55 
9  0  23 
9  19  29 
9  38  21 
9  56  60 
10  15  27 
10  33  44 

h.  min.  sec. 

10  41  0 
10  55  29 
11  13  30 
11  31  28 
11  49  25 
12  7  24 
12  25  27 

h.  mi  >.  stc. 

12  29  4 
12  43  36 
13  1  54 
13  20  24 
13  39  8 
13  58  9 
14  17  27 

h.  min.  sec. 

14  25  16 
14  41  2 
15  1  5 
15  21  28 
15  42  14 
16  3  19 
16  24  43 

h.  rain.  sec. 

16  29  1 
16  46  23 
17  8  17 
17  30  22 
17  52  33 
18  14  46 
18  36  57 

The  R.  A.  in  this  table  will  answer  for  corresponding  days,  in  other  years* 
within  four  minutes  ;  and  for  periods  of  four  years,  the  difference  is  only  about 
•ere  a  M«ands  for  each  period. 


TABLE   III. 


TABULAR    VIEW    OF    THE    SOLAR    SYSTEM. 


Names. 

Mean  diameters  in 
miles. 

Mean  distance  Mean  dist.;|    Log.  of   (Time  of  revolu- 
from  the  Sun  the  Earth's      mean     1     tions  round 
in  mile?.        dist.    unity.]  distance.             Sun. 

Log.  of  ' 
times  of 
revolution 

1.944324 
2.351610 
2  562598 
2.836942 
3.121991 
3.123190 
3.138303 
3.167300 
3.179547 
3.202700 
3.226086 
3.226610 
3.636738 
4.03171S 
4.486953 
4.779076 

Sun  

Mercury  . 
Venus  ..  . 
The  Earth 
Mars 
Vesta  .  .  . 
Iris       1  . 
Hebe     1 
Flora     f* 
AstreaJ  . 
Juno 

883000 
3224 
7687 
7912 
4189 
238 

>  Unknown. 

1420 

Not  well   Q  60 
known.   Jl20 
89170 
79040 
35000 
35000 

37  million 
68      " 
95      " 
144       " 
224,340,000 
226  raiilion 
230       " 
240       « 
246      " 
253,600,001) 
263,236,000 
265  million 
490       " 
900       " 
1800       « 
2850      " 

0.387098 
0.723332 
1.000000 
1.52369-2 
2.36120 
2.37880 
2.42190 
252630 
2.5895 
2.66514 
2.76910 
2.77125 
5.202776 
9.538786 
19182390 
29.59 

DAYS. 

9.5878181      87.969258 
9.859306     224.700787 
0.0(10000,     365.256383 
0.182810     686979646 
0.373100   1324.289 
0.376384   1327.973 
0384104   1375.  nearly 
0.402487   1469.76 
0.413211    I5l2.nearly 
0.425710    1594.721    " 
0442334   1683.064 
0.442725    1685162 
0.716212  4332.584821 
097947610759,219817 
1  .282853  30686.8208 
1.477121,60128  14 

Ceres  
Pallas  .  .  . 
Jupiter..  . 
Saturn  .  .  . 
Uranus  .  . 
Neptune  . 

TABLE   III. 


ELEMENTS  OF  ORBITS    FOR    THE    EPOCH    OF  1850,  JANUARY  1,  MEAN    NOON   AT 

GREENWICH. 


Planets. 

Inclinati'u 
of  orbits 
to  ecliptic. 

Variation 
in  100 
years. 

Long.  of  the 
ascending 
nodes. 

Variation 
in  100 
years. 

Longitude 
of 
Perihelion. 

Variation 
in  100 

years. 

Mean  longi- 
tude  at 
epoch. 

Mercury 
Venus.  . 
Earth  .  .  . 
Mars  .  .  . 
Vesta... 

O      '    " 
7     0   18 
3  23  26 

1  51     6 
7     8  29 
13    2  53 

H-18.2 
—  4.6 

—  0.2 
—12. 

O     '     " 
46  34  40 
75  17  40 

48  20  24 
103  20  47 
170  53    0 

+51 

+42 
+26 

O       '      " 

75    9  47 
129  22  53 
100  22  10 
333  17  57 
2.74    4  34 
54  18  32 

+  93 
+  78 
103 
+110 
157 

C              ' 

327  17     9 
243  58    4 
100  47     1 
182    9  30 
113  28  12 
165  17  38 

10  37  17 

80  47  56 

147  25  41 

1     3  10 

Pallas    . 

34  37  44 

172  42  38 

121  30  13 

327  31  24 

Jupiter.. 
Saturn.  . 
Uranus.. 

1  18  42 
2  29  29 
0  46  27 

—22. 
—15. 
3 

98  55  19 
112  22  54 
73  12    0 

+57 

+-51 
+24 

11   56    0 
90    7     0 
168  14  47 

+  95 
+116 

+  87 

160  21  50 
13  58  13 
28  20  22 

*  Recently  some  thirty-two  Astero  Js  have  been  discovered,  by  different  «iD- 
•ervers.  A  table  of  twenty-two  wi\  be  found  on  page  55  of  tables.  We  give 
the  logarithms  in  the  tables,  that  the  data  may  be  at  hand  to  exercise  the  stu- 
dent cu  Kepler's  third  law. 


TABLE   III. 


TABULAR    VIEW    OF    THE    SOLAR    SYSTEM. 


Names. 

Mass. 

Density. 

Gravity. 

Siderial. 
Rotation. 

Light  and 
Heat. 

Mercury  .  . 

WlllTV 

3.244 

1.22 

b.       m.         i. 

24      5    28 

6.680 

Venus 
Earth  

0.994 
1  000 

0.96 
1.00 

23    21      7 
24      0      0 

1.911 
1  000 

Mare  

ysstan 

0.973 

0.50 

24    39    21 

.431 

Jupiter  .  .  . 

TffiffT 

0.232 

2.70 

9    55    50 

.037 

Saturn  

ssiff.y 

0.132 

1.25 

10    29    17 

.011 

Uranus  .  .  . 

T^iif 

0.246 

1.06 

Unknown. 

.003 

Sun  
Moon  

1 

0.256 
0.665 

28.19 
0.18 

25d.  12h.   Om. 

27*.    7h.43m. 

TABLE   III. 


Planets. 

Eccentricities 
of  orbits. 

Variation  in  100 
years. 

Motion  in  mean 
long,  in  1  year 
of  365  days. 

Mean  Daily 
Motion  in 
longitude. 

Mercury... 
Venus  
Earth  

0.20551494 
0.00686074 
0  01678357 

+  .000003868 
—  .000062711 
—  .000041630 

0       '       " 
53  43    3.6 
224  47  29.7 
—0  14  19.5 

O       '     " 

4    5  32.6 
1  36    7.8 
0  59    8.3 

Mars  

0  09330700 

+  .000090176 

191  17    9.1 

0  31  26.7 

Vesta 

0  08856000 

4-  000004009 

0  16  179 

Juno  .  • 

0  25556000 

0  13  33  7 

Ceres     . 

0  07673780 

—  000005830 

0  12  494 

Pallas 

024199800 

0  12  487 

Jupiter  
Saturn  
Uranus  

0.04816210 
0.05615050 
0.04661080 

-+-  .000159350 
—  .000312402 
—  .000025072 

30  20  31.9 
12  13  36.1 
4  17  45.1 

0    4  59.3 
0    2    0.6 
0    U  42.4 

TABLE     III. 

LUNAR     PERIODS. 


d. 


Mean  sidereal  revolution, 27.321661418 

Mean  synodical  revolution, 29.530588715 

Mean  revolution  of  nodes  (retrograde), 6793.391080 

Mean  revolution  of  perigee  (direct),   3232.575343 

Mean  inclination  of  orbit, 5°  8'  48" 

Mean  distance,  in  measure,  of  the  equc*orial  radius  of 

the  earth, .. 29.98217 

Mean  distance,  in  measure,  of  the  mean  rudius, 30.20000 


TABLE    IV 
SUN'S  EPOCHS. 


Yeara. 

M.  Long. 

Long.  Perigee. 

I. 

II. 

III. 

N. 

1.  0   '    " 

8.   0    '    " 

1846 

9  8  45   8 

9  8  17  17 

124 

673 

897 

379 

1847 

9  8  30  48 

9  8  18  19 

484 

588 

623 

433 

1848  B. 

9  9  15  37 

9  8  19  20 

878 

505 

151 

487 

1849 

99   1  17 

9  8  20  22 

2.S8 

420 

775 

540 

1850 

9  8  46  58 

9  8  21  23 

598 

336 

400 

594 

1851 

9  8  32  39 

9  8  22  24 

958 

250 

025 

648 

1852  B. 

9  9  17  27 

9  8  23  26 

353 

168 

653 

701 

1853 

9938 

9  8  24  27 

713 

083 

277 

755 

1854 

9  8  48  48 

9  8  25  29 

073 

998 

902 

809 

1855 

9  8  34  29 

9  8  26  30 

433 

913 

527 

863 

1856  B. 

9  9  19  18 

9  8  27  32 

827 

832 

153 

916 

1857 

9  9   4  58 

9  8  28  34 

187 

746 

779 

970 

1858 

9  8  50  39 

9  8  29  35 

547 

661 

404 

024 

1859 

9  8  36  19 

9  8  30  37 

907 

576 

029 

078 

1860  B 

9  9  21   8 

9  8  31  38 

301 

494 

656 

131 

1861 

9  9   6  49 

9  8  32  39 

661 

409 

281 

185 

1862 

9  8  52  29 

9  8  33  41 

021 

324 

906 

239 

1863 

9  8  38  10 

9  8  34  42 

381 

239 

530 

292 

1864  B. 

9  9  22  58 

9  8  35  44 

775 

157 

157 

346 

1865 

9  9   8  39 

9  8  36  45 

135 

072 

783 

400 

1866 

9  8  54  20 

9  8  37  47 

495 

985 

408 

453 

1867 

9  8  40   0 

9  8  38  49 

855 

902 

033 

507 

1868  B. 

9  9  24  49 

9  8  39  50 

249 

820 

659 

561 

1869 

9  9  10  30 

9  8  40  52 

609 

734 

285 

615 

1870 

9  8  56  10 

9  8  41  53 

969 

649 

910 

668 

1882 

9  9   1  41 

9  8  54  10 

391 

638 

416 

313 

1871 

9  8  41  51 

9  8  42  54 

329 

564 

534 

721 

1872  B 

9  9  26  39 

9  8  43  56 

723 

481 

161 

774 

1873 

9  9  12  20 

9  8  45  58 

083 

396 

785 

828 

1874 

9  8  58   1 

9  8  47   0 

443 

311 

410 

881 

1875 

9  8  43  41 

9  8  48   2 

803 

226 

034 

935 

1876  B. 

9  9  28  30 

9  8  49   4 

297 

143 

661 

989 

1877 

9  9  14  10 

9  8  50   5 

657 

058 

286 

042 

1878 

9  8  59  51 

9  8  51   6 

017 

974 

912 

096 

1879 

9  8  45  32 

9  8  52   7 

377 

889 

537 

150 

IbSOB. 

9  9  30  20 

9  8  53   9 

671 

807 

164 

204 

1881 

9  9  16   1 

9  8  54  10 

031 

722 

790 

257 

1882 

9  9   1  41 

9  8  55  12 

391 

637 

415 

311 

1883 

9  8  47  22 

9  8  56  13 

751 

552 

040 

364 

1884  B. 

9  9  32  10 

9  8  57  15 

145 

469 

666 

418 

1885 

9  9  17  51 

9  8  58  16 

505 

385 

292 

471 

1886 

9  9   3  32 

9  8  59  17 

865 

300 

918 

525 

1887 

9  8  49  12 

9  8   0  19 

225 

216 

544 

579 

1888  B. 

9  9  34   1 

9  8   1  20 

619 

133 

169 

632 

10 


TABLE    V. 
SUN'S   MOTIONS   FOR   MONTHS. 


Months. 

Longitude. 

Per. 

I. 

II. 

III. 

N. 

T       1  Com.  .  .  . 
Jan-jBis  
r*  .    I  Com.  .  .  . 
Feb'jBis  
March 

s.      o      '      " 
0000 
11     29      0    52 
1       0    33     18 
0    29    34    10 
1     28      9    11 

0 
0 
5 
5 
10 

0 
966 
47 
13 
993 

« 

0 
997 
78 
7fi 
148 

0 
998 
53 
51 
01 

0 
0 
4 
4 
9 

April 

2    28    42    30 

15 

42 

226 

154 

13 

May  

3    28    16    40 

20 

59 

301 

206 

18 

4    28    49    58 

26 

110 

379 

259 

22 

July    . 

5    28    24      8 

31 

129 

454 

310 

27 

6    28    57    26 

36 

182 

531 

363 

31 

September.  •  .  . 

7    29    30    44 

41 

233 

609 

416 

36 

October.  .  . 

8    29      4    54 

46 

250 

684 

468 

40 

Novembf  r  . 

9    29    38    12 

52 

300 

762 

521 

45 

10    29     12    22 

57 

313 

837 

572 

49 

TABLE    VI. 

SUN'S    HOURLY   MOTION. 
ARGUMENT. — Sun's  Mean  Anomaly. 


Os 

Is 

Us 

Ills 

IVs 

2    25 
2    25 
2    24 
2    24 

V 

0      . 

0 
10 
20 
30 

'      n 
2    33 
2    33 
2    33 
2    32 

2    32 
2    32 
2    31 
2    30 

/     n 

2    30 
2    29 
2    29 
2    28 

2    28 
2    27 
2    26 
2    25 

2    24 
2    23 
2    23 
2    23 

0 

30 
20 
10 
0 

XIs 

Xs 

IXs 

VIIIs 

VIIs 

Vis 

SUN'S   SEMIDIAMETER. 
ARGUMENT. — Sun's  Mean  Anomaly. 


Os 

Is 

Us 

Ills 

IVs 

VB 

o 

»      a 

/      // 

/     / 

/        // 

/       // 

i       n 

o 

0 

16    18 

16    15 

16    9 

16      1 

15    53 

15    48 

30 

10 

16    18 

16    14 

16    7 

15    58 

15    51 

15    46 

20 

20 

16    17 

16    12 

16    4 

15    56 

15    49 

15    46 

10 

30 

16    15 

16      9 

16    1 

15    53 

15    48 

15    45 

0 

XIj 

Xs 

IXs 

VIIIs 

vn* 

VI« 

TABLE    VII. 

SUN'S   MOTIONS   FOE   DAYS   AND   IIOUBS. 


11 


j  Days. 

Logitude. 

Per 

I. 

II. 

III 

N. 

Hours. 

Long. 

I. 

O   i    » 

a 

,   »» 

1 

000 

0 

0 

0 

0 

0 

1 

2  28 

1 

2 

0  59   8 

0 

34 

3 

2 

0 

2 

4  56 

3 

3 

1  58  17 

0 

68 

5 

3 

0 

3 

7  23 

4 

4 

2  57  25 

0 

101 

8 

5 

0 

4 

9  51 

ti 

5 

3  56  33 

1 

135 

10 

7 

1 

5 

12  19 

7 

6 

4  55  42 

169 

13 

9 

1 

6 

14  47 

8 

7 

5  54  50 

203 

15 

10 

1 

7 

17  15 

10 

8 

6  53  58 

236 

18 

12 

1 

8 

19  43 

11 

9 

7  53   7 

270 

20 

14 

9 

22  11 

13 

10 

8  52  15 

304 

23 

15 

1 

10 

24  38 

14 

11 

9  51  23 

2 

338 

25 

17 

1 

11 

27  6 

16 

12 

10  50  32 

2 

371 

28 

19 

2 

12 

29  34 

17 

13 

11  49  40 

2 

405 

30 

21 

2 

13 

32  2 

18 

14 

12  48  48 

2 

439 

33 

22 

2 

14 

34  30 

20 

15 

13  47  57 

2 

473 

35 

24 

2 

15 

36  58 

21 

16 

14  47   5 

3 

506 

38 

26 

2 

16 

39  26 

23 

17 

15  46  13 

3 

540 

40 

27 

2 

17 

41  53 

24 

13 

16  45  22 

3 

574 

43 

29 

2 

18 

44  21 

25 

19 

17  44  30 

3 

608 

45 

31 

3 

19 

46  49 

27 

20 

18  43  38 

3 

641 

48 

33 

3 

20 

49  17 

28 

21 

19  42  47 

3 

675 

50 

34 

3 

21 

51  45 

30 

22 

20  41  55 

4 

709 

53 

36 

3 

22 

54  13 

31 

23 

21  41   3 

4 

743 

55 

38 

3 

23 

56  40 

32 

24 

22  40  12 

4 

777 

58 

39 

3 

24 

59  8 

34 

25 

23  39  20 

4 

810 

60 

41 

4 

26 

24  38  28 

4 

844 

63 

43 

4 

27 

25  37  37 

4 

878 

65 

45 

4 

23 

26  36  45 

5 

912 

68 

46 

4 

29 

27  35  53 

5 

945 

70 

48 

4 

30 

28  35   2 

5 

979 

73 

50 

4 

31 

29  34  10 

5 

13 

75 

51 

4 

SONS   MOTIONS   FOR   MINUTES. 


Min. 

Longitude. 

Min. 

Longitude. 

1 
5 

10 
15 
20 
25 
30 

0        2 
0       12 
0      25 
0      37 
0      49 
1        2 
1       14 

30 
35 
40 
45 
50 
55 
6Q 

/        a 

1      16 
1      26 

1      39 
1      51 
2        3 
2      16 

2      28 

12 


TABLE    VIII. 


EQUATIONS   OP   THE    SUN^S    CENTER. 
ARGUMENT. — Sun's  Mean  Anomaly. 


Os 

Is 

Us 

Ills 

IV. 

Vs 

o 

O   '   " 

Q   t    II 

O   '    " 

Q    t    II 

o  •   » 

O   '   n 

0 

1  59  30 

2  58  15 

3  40  27 

3  54  50 

3  38  21 

2  56  9 

1 

2  1  33 

300 

3  41  25 

3  54  47 

3  37  18 

2  54  25 

2 

2  3  37 

3  1  44 

3  42  21 

3  54  41 

3  36  14 

2  52  40 

3 

2  5  40 

3  3  27 

3  43  15 

3  54  33 

3  35  8 

2  50  54 

4 

2  7  43 

359 

3  44  8 

3  54  23 

3  34  1 

2  49  8 

5 

2  9  46 

3  6  49 

3  44  58 

3  54  11 

3  32  51  . 

2  47  20 

6 

2  11  49 

3  8  28 

3  45  47 

3  53  57 

3  31  41 

2  45  32 

7 

2  13  51 

3  10  6 

3  46  33 

3  53  41 

3  30  28 

2  43  43 

8 

2  15  54 

3  11  43 

3  47  17 

3  53  23 

3  29  14 

2  41  53 

9 

2  17  56 

3  13  18 

3  48  0 

3  53  3 

3  27  58 

2  40  3 

10 

2  19  57 

3  14  51 

3  48  40 

3  52  40 

3  26  41 

2  38  11 

11 

2  21  58 

3  16  24 

3  49  18 

3  52  16 

3  25  22 

2  36  19 

12 

2  23  59 

3  17  54 

3  49  55 

3  51  50 

3  24  2 

2  34  27 

13 

2  25  59 

2  19  24 

3  50  29 

3  51  21 

3  22  40 

2  32  34 

14 

2  27  59 

3  20  51 

3  51   1 

3  50  51 

3  21  17 

2  30  40 

15 

2  29  58 

3  22  18 

3  51  31 

3  50  18 

3  19  52 

2  28  46 

16 

2  31  57 

3  23  42 

3  51  59 

3  49  44 

3  18  26 

2  26  52 

17 

2  33  55 

3  25  5 

3  52  25 

3  49  7 

3  16  58 

2  24  56 

18 

2  35  52 

3  26  26 

3  52  49 

3  48  29 

3  15  30 

2  23  0 

19 

2  37  49 

3  27  46 

3  53  10 

3  47  49 

3  14  0 

2  21  4 

30 

2  39  45 

3  29  4 

3  53  30 

3  47  7 

3  12  28 

2  19  8 

21 

2  41  40 

3  30  24 

3  53  47 

3  46  22 

3  10  55 

2  17  11 

22 

2  43  34 

3  31  35 

3  54  3 

3  45  36 

3  9  22 

2  15  14 

23 

2  45  28 

3  32  48 

3  54  16 

3  44  48 

3  7  46 

2  13  16 

24 

2  47  20 

3  33  59 

3  54  27 

3  43  58 

3  6  10 

2  11  19 

25 

2  49  12 

3  35  8 

3  54  36 

3  43  7 

3  4  33 

2  9  21 

26 

2  51  2 

3  36  16 

3  54  43 

3  42  13 

3  2  54 

2  7  23 

27 

2  52  52 

3  37  21 

3  54  48 

3  41  18 

3  1  14 

2  5  25 

28 

2  54  41 

3  38  25 

3  54  51 

3  40  21 

2  59  33 

2  3  27 

29 

2  56  28 

3  39  27 

3  54  52 

3  39  22 

2  57  52 

2  1  25 

30 

2  58  15 

3  40  37 

3  54  50 

3  38  91 

2  56  9 

1  59  30  i 

TABLE    VIII. 


13 


EQUATIONS   OF   THE    SUN  8   CENTER. 
ARGUMENT. — Sun's  Mean  Anomaly. 


Vis 

VIIs 

VIIIs 

IXs 

Xs 

XIs 

o 

o     '     " 

O      '         " 

O      '        " 

O        '         " 

0       '        " 

Q         1          II 

0 

1    59    30 

1     2   51 

0    20    39 

0     4   10 

0    18    33 

0  45 

1 

1    57    32 

1     1     8 

0    19    38 

048 

0    19    33 

2   32 

2 

1    55    33 

0   59    27 

0    18    39 

049 

0   20    35 

4   19 

3 

1    53    35 

0    57   46 

0    17    42 

0     4   12 

0   21    39 

6     8 

4 

1    51    37 

0   56     6 

0    16   47 

0     4   17 

0   22   44 

7   58 

5 

1    49    39 

0   54   27 

0    15    53 

0     4  24 

0   23    52 

9   48 

6 

1    47    41 

0   52   47 

0    15     2 

0     4   33 

0   25     1 

11   40 

7 

1    45    44 

0   51    14 

0    14    12 

0     4   44 

0   26    12 

13   32 

8 

1    43    46 

0   49    38 

0    13    24 

0     4   57 

0   27    25 

15   26 

9 

1    41    49 

0   48     5 

0    12    38 

0     5    13 

0    28   40 

17   20 

10 

1    39    52 

0   46   32 

0    11    53 

0     5   30 

0   29   56 

19    15 

11 

1    37    56 

0   45     0 

0    11    11 

0      5    50 

0    31    14 

21    11 

12 

1    36     0 

0   43    30 

0    10    31 

0      6    11 

0    32    34 

23     8 

13 

34     4 

0    42      1 

0     9    53 

0     6   35 

0    33    55 

25     5 

14 

32     9 

0    40    34 

0     9    16 

0     7     1 

0    35    18 

27     3 

15 

30    14 

0    39      8 

0     8    42 

0     7   29 

0    36   42 

29     2 

16 

28    20 

0    37    43 

089 

0     7    59 

0    38     9 

1    31      1 

17 

26   26 

0    36   20 

0     7    39 

0     8   31 

0    39    36 

1    33     1 

18 

24    33 

0    34   58 

0     7    10 

095 

0   41     9 

1    35     1 

19 

22   41 

0    33    38 

0     6   44 

0     9   42 

0   42   36 

1    37     1 

20 

20   49 

0    32   19 

0     6   20 

0    10   20 

0   44     9 

1    39     3 

21 

18    57 

0    31     2 

0     5    57 

0    11      0 

0   45   42 

41     4 

22 

17     7 

0    29    46 

0     5    37 

0    11    43 

0   47    17 

43     6 

23 

15    17 

0   28   32 

0     5    19 

0    12   27 

0   48   54 

45     9 

24 

13    28 

0    27    19 

053 

0    13    13 

0   50   32 

47    11 

25 

11    40 

0   26     9 

0     4   49 

0    14     2 

0   52   11 

49    14 

26 

9    52 

0   24   59 

0     4   37 

0    14   52 

0   53   51 

51    17 

27 

8     6 

0   23   52 

0     4   27 

0    15   45 

0   55   33 

53   SO 

28 

6   20 

0   22   46 

0     4   19 

0    16   39 

0   57    16 

1    55   23 

39 

4   35 

0   21    41 

0     4   13 

0    17    35 

0   59     0 

1    57   27 

30 

1      2   51 

0   20   39 

0     4   10 

0    18  33 

1     0   45 

1    59   30 

TABLE    IX. 

SMALL   EQUATIONS   OF   THE   SUN*S   LONGITUDE 


Arg. 

I 

II. 

III. 

Arg. 

I. 

II. 

III 

0 

10 

10 

II 

10 

500 

10 

10 

10 

10 

10 

11 

9 

510 

10 

10 

9 

20 

11 

11 

9 

520 

9 

10 

8 

30 

11 

12 

8 

530 

9 

10 

7 

40 

11 

13 

8 

540 

9 

10 

7 

50 

12 

14 

7 

550 

8 

10 

6 

60 

12 

14 

7 

563 

8 

9 

5 

70 

12 

15 

7 

570 

8 

9 

4 

80 

13 

15 

7 

580 

7 

9 

3 

90 

13 

16 

7 

590 

7 

9 

3 

103 

13 

16 

7 

600 

7 

9 

2 

110 

14 

17 

7 

610 

6 

8 

1 

120 

14 

17 

7 

620 

6 

8 

1 

130 

14 

18 

8 

630 

6 

8 

1 

140 

15 

18 

8 

640 

5 

7 

0 

150 

15 

18 

9 

650 

5 

7 

C 

160 

15 

18 

9 

660 

5 

6 

0 

170 

15 

18 

10 

670 

5 

6 

1 

180 

15 

18 

10 

680 

5 

6 

1 

190 

16 

18 

11 

690 

4 

5 

2 

2)0 

16 

18 

11 

700 

4 

5 

2 

210 

16 

18 

i2 

710 

4 

4 

3 

220 

16 

18 

12 

720 

4 

4 

3 

230 

16 

18 

13 

730 

4 

4 

4 

240 

16 

17 

14 

740 

4 

3 

5 

250 

16 

17 

14 

750 

4 

3 

6 

260 

16 

17 

15 

760 

4 

3 

6 

270 

16 

16 

16 

770 

4 

2 

7 

280 

16 

16 

17 

7fO 

4 

2 

8 

290 

16 

16 

17 

790 

4 

2 

8 

300 

16 

15 

18 

800 

4 

2 

9 

310 

16 

15 

18 

810 

4 

2 

9 

320 

15 

14 

19 

820 

5 

2 

10 

330 

15 

14 

19 

830 

5 

2 

10 

340 

15 

14 

20 

840 

5 

2 

11 

350 

15 

13 

20 

850 

5 

2 

11 

360 

15 

13 

20 

860 

5 

2 

12 

370 

14 

12 

19 

H70 

6 

2 

12 

380 

14 

12 

19 

880 

6 

3 

13 

390 

14 

12 

19 

8!K) 

6 

3 

13 

400 

13 

11 

18 

900 

7 

4 

13 

410 

13 

11 

17 

910 

7 

4 

13 

420 

13 

11 

17 

920 

7 

5 

13 

430 

12 

11 

16 

930 

8 

5 

13 

440 

12 

11 

15 

940 

P 

6 

13 

450 

12 

10 

14 

950 

8 

6 

13 

463 

11 

10 

13 

96!) 

9 

7 

12 

470 

11 

10 

13 

970 

9 

8 

12 

480 

11 

10 

12 

983 

9 

9 

11 

490 

10 

10 

11 

990 

10 

£ 

11 

500 

10 

10 

10 

1000 

10 

16 

10 

TABLE    X 

NUTATIONS. 
ARGUMENT. — Supplement  of  the  Node,  or  N. 


15 


N. 

Long. 

R.  Asc. 

Obliq. 

N. 

Long. 

R.  Asc. 

Obliq. 

„ 

a 

pf 

n 

„ 

„ 

0 

4-  o 

4-  ° 

4-  10 

500 

—  0 

—  0 

—  10 

20 

2 

2 

10 

520 

2 

2 

9 

40 

4 

4 

9 

540 

4 

4 

9 

60 

7 

6 

9 

560 

7 

6 

9 

80 

9 

8 

8 

580 

9 

8 

8 

100 

I  jo 

600 

—  11 

—  10 

—  8 

120 

12 

11 

7 

620 

12 

11 

7 

140 

14 

13 

6 

640 

14 

13 

6 

160 

15 

14 

5 

660 

15 

14 

5 

180 

16 

15 

4 

680 

16 

15 

4 

200 

4-  17 

4-  16 

t   3 

700 

—  17 

—  16 

—  3 

220 

r!8 

16 

2 

720 

18 

16 

2 

240 

18 

16 

1 

740 

18 

16 

1 

260 

18 

16 

1 

760 

18 

16 

+  * 

280 

18 

16 

2 

780 

18 

16 

2 

300 

4-  17 

4-  16 

_  3 

800 

—  17 

—  16 

4"  3 

320 

16 

^*5 

4 

820 

16 

15 

4 

340 

15 

14 

5 

840 

15 

14 

5 

366 

14 

13 

6 

860 

14 

13 

6 

380 

12 

11 

7 

880 

12 

11 

7 

400 

4.  10 

—  8 

900 

—  11 

—  10 

+  8 

420 

9 

~  8 

8 

920 

9 

8 

8 

440 

7 

6 

9 

940 

7 

6 

9 

460 

4 

4 

9 

960 

4 

4 

9 

480 

2 

2 

10 

980 

2 

2 

10 

500 

~f"  0 

—  10 

1000 

—  0 

—  0 

£15 

TABLE    XI. 
EARTH'S  RADIUS  VECTOR. — ARGUMENT.  Sun's  Mean  Anomaly. 


Os 

Is 

Us 

Ills 

IVs 

V« 

GO 

0.98313 

0.98545 

0.99173 

.00018 

1.00850 

1.01450 

300 

2 

0.98314 

0.98576 

0.99225 

1.00077 

1.00899 

1.01477 

28 

4 

0.98317 

0.98608 

0.99278 

1.00135 

1/0947 

1.01503 

26 

6 

0.98322 

0.98643 

0.99331 

.10193 

1.00994 

1.01527 

24 

8 

0.98330 

0.98679 

0.99386 

1.00251 

1.01040 

1.01549 

22 

10 

0.98339 

0.98717 

0.99441 

.00308 

1.01084 

1.01569 

20 

12 

0.98350 

0.98756 

0.99497 

.00366 

1.01128 

1.01588 

18 

14 

0.98364 

0.98797 

0.99554 

.00422 

1.01170 

1.01604 

16 

16 

0.98380 

0.98840 

0.99611 

.00478 

1.01210 

1.01619 

14 

18 

0.98397 

0.98883 

0.99668 

.00534 

1.01249 

1.01632 

12 

20 

0.98417 

0.98929 

0.99726 

1.00588 

1.01286 

1.01643 

10 

22 

0.98439 

0.98975 

0.99784 

1.00642 

1.01322 

1.01652 

8 

24 

0.98462 

0.99023 

0.99843 

1.00695 

1.01357 

1.01659 

6 

26 

0.98486 

0.99072 

0.99901 

1.00748 

1.01389 

1.01663 

4 

28 

0.98515 

0.99122 

0.99960 

1.00799 

1.01420 

1.01666 

2 

30 

0.98545 

0.99173 

1.00018 

1.00850 

1.01450 

1.01667 

0 

XIi 

Xs 

IXs 

vim 

vm 

VI. 

TABLE  XI. 


MEAN    NEW    MOONS    AND    ARGUMENTS    IN    JANUARY. 


Mean  New 
Moon  in 
January. 

I. 

II. 

in. 

IV. 

N. 

A  D 

D,  H.  M. 

1836  B. 

17  10  32 

0469 

1246 

17 

08 

669 

1837 

5  19  20 

0171 

9S52 

00 

97 

692 

1838 

24  16  53 

0681 

9175 

99 

85 

799 

1839 

14  1  42 

0383 

7780 

82 

74 

822 

1840  B. 

3  10  30 

0085 

6386 

65 

63 

844 

1841, 

21  8  3 

0595 

5709 

63 

51 

951 

1842 

10  16  51 

0297 

4314 

46 

40 

974 

1843 

29  14  24 

0807 

3637 

44 

28 

081 

1844  B. 

18  23  13 

0509 

2243 

28 

17 

104 

1845 

7  8  1 

0211 

0848 

11 

06 

126 

1846 

26  5  34 

0721 

0171 

09 

94 

234 

1847 

15  14  22 

0423 

8777 

92 

84 

256 

1848  B. 

4  23  11 

01-25 

7382 

75 

73 

278 

1849 

22  20  43 

0635 

6705 

73 

61 

386 

1850 

12  5  32 

0337 

53il 

56 

50 

408 

1851 

1  14  21 

0038 

3916 

40 

39 

431 

1852  B. 

20  11  53 

0549 

3239 

38 

27 

538 

1853 

8  20  42 

0251 

1845 

21 

16 

560 

1854 

27  18  14 

0761 

1168 

19 

04 

668 

1855 

17  3  3 

0463 

9773 

02 

93 

690 

1856  B 

6  11  51 

0164 

8379 

85 

82 

713 

1857 

24  9  24 

0675 

7702 

84 

70 

820 

1858 

13  18  13 

0376 

6307 

67 

59 

843 

1859 

3  3  1 

0078 

4913 

50 

48 

865 

1860  B. 

22  0  34 

0588 

4236 

48 

36 

972 

1861 

10  9  22 

0290 

2840 

31 

25 

995 

1862 

29  6  55 

0800 

2163 

30 

14 

102 

1863 

18  15  44 

0504 

0769 

13 

03 

125 

1864  B. 

8  0  32 

0204 

9374 

96 

92 

147 

1865 

25  22  5 

0714 

8698 

94 

80 

256 

1866 

15  6  53 

0416 

7303 

77 

69 

277 

1867 

4  15  42 

0118 

5909 

60 

58 

299 

1868  B. 

23  13  14 

0628 

5231 

59 

46 

407 

1869 

11  22  3 

0330 

3837 

42 

35 

429 

1870 

1  6  51 

0032 

2442 

25 

24 

451 

1871 

20  4  24 

0542 

1765 

23 

12 

559 

1872  B. 

8  13  13 

0244 

0371 

05 

01 

581 

1873 

27  10  46 

0754 

9694 

03 

89 

689, 

1874 

17  19  35 

0456 

8300 

86 

78 

711 

1875 

7  4  24 

0158 

6906 

69 

67 

733 

1876  B. 

26  1  57 

0668 

6229 

67 

55 

841 

1877 

14  10  49 

0370 

4835 

50 

44 

863 

1678 

3  18  38 

0072 

3441 

33 

23 

885 

1879 

22  6  11 

0582 

2764 

31 

21 

993 

1880  B. 

il  15  0 

0284 

1370 

14 

10 

015 

TABLE    XII. 


IT 


MEAN    LUNATIONS    AND    CHANGES    OF   THE   ARGUMENTS. 


Num. 

Lunations. 

I. 

II. 

III. 

IV. 

N. 

d.   h.  m. 

i/ 

14  18  22 

404 

5359 

58 

50 

43 

j 

29  12  44 

808 

717 

15 

99 

85 

2 

59   1  28 

1617 

1434 

31 

98 

170 

3 

88  14  12 

2425 

2151 

46 

97 

256 

4 

118   2  56 

3234 

2869 

61 

96 

341 

5 

147  15  40 

4042 

3586 

76 

95 

425 

6 

177   4  24 

4851 

4303 

92 

95 

511 

7 

206  17   8 

5659 

5020 

7 

94 

596 

8 

236   5  52 

6468 

5737 

22 

93 

682 

9 

265  18  36 

7276 

6454 

37 

92 

767 

10 

295   7  20 

8085 

7171 

53 

91 

852 

11 

324  20   5 

8893 

7889 

68 

90 

937 

IS 

354   8  49 

9702 

8606 

83 

89 

22 

U 

383  21  33 

510 

9323 

93 

88 

108 

TABLE    XIII. 

KCMBER  OP  DAYS  FROM  THE 
COMMENCEMENT  OF  THE  YEAR 
TO  THE  FIRST  OF  EACH  MONTH. 


TABLE  XIV. 


Months. 

Com. 

Bis. 

Days. 

Days. 

January,  .  . 

0 

0 

February.  . 

31 

31 

March  .... 

59 

60 

April  

90 

91 

May.  .  , 

120 

121 

151 

152 

July  .  . 

181 

182 

August..  .  . 

212 

213 

September. 

243 

244 

October... 

273 

274 

November. 

304 

305 

December  . 

334 

335 

Aff 

• 
H.  Par. 

m 

S.D. 

» 

H.Mo. 

AiF 

i   " 

/   it 

0 

60  29 

16  29 

36  48 

10000 

250 

60  26 

16  26 

36  44 

9750 

500 

60  17 

16  25 

36  19 

9500 

750 

60   0 

16  21 

36   8 

9250 

1000 

59  47 

16  17 

35  48 

9000 

1250 

59  24 

16  11 

35  28 

«750 

1500 

58  56 

16   3 

34  57 

8500 

1750 

58  30 

15  56 

34  34 

8250 

2000 

58   7 

15  50 

33  58 

8000 

2250 

57  37 

15  42 

33  32 

7750 

2500 

57   1 

15  31 

32  42 

7500 

2750 

56  32 

15  23 

32   9 

7250 

3000 

56   2 

15  16 

31  36 

7000 

3250 

55  40 

15  10 

31  13 

6750 

3500 

55  22 

15   7 

30  52 

6500 

3750 

55  12 

15   3 

30  29 

6250 

4000 

54  51 

14  56 

30   7 

6000 

4250 

54  39 

14  54 

29  55 

5750 

4500 

54  26 

14  50 

29  43 

5500 

4750 

54  18 

14  48 

29  37 

5250 

5000 

54  13 

14  45 

29  35 

5000 

18 


TABLE   XV. 

EQUATIONS   FOR   NEW   AND    FULL   MOON. 


Arg. 

I. 

II. 

Arg. 

1. 

II. 

Arg. 

III. 

IV. 

Arg. 

h.  m. 

h.  m. 

h.  m. 

h.  m. 

m. 

m. 

0 

4  20 

10  10 

5000 

4  20 

10  10 

25 

3 

31 

25 

100 

4  36 

9  36 

5100 

4  5 

10  50 

26 

3 

31 

24 

200 

4  52 

9  2 

5200 

3  49 

11  30 

27 

3 

30 

23 

300 

5  8 

8  28 

5300 

3  34 

12  9 

28 

3 

30 

22 

400 

5  24 

7  55 

5400 

3  19 

12  48 

29 

3 

30 

21 

500 

5  40 

7  22 

5500 

3  4 

13  26 

30 

3 

30 

20 

600 

5  55 

6  49 

5600 

2  49 

14  3 

31 

3 

30 

19 

700 

6  10 

6  17 

5700 

2  35 

14  39 

32 

4 

20 

18 

800 

6  24 

5  46 

5800 

2  21 

15  13 

33 

4 

29 

17 

900 

6  38 

5  15 

5900 

2  8 

15  46 

34 

4 

29 

16 

1000 

6  51 

4  46 

6000 

55 

16  18 

35 

4 

29 

15 

1100 

7  4 

4  17 

6100 

42 

16  48 

36 

5 

28 

14 

1200 

7  15 

3  50 

6200 

31 

17  16 

37 

5 

28 

13 

1300 

7  27 

3  24 

6300 

19 

17  42 

38 

5 

27 

12 

uoa 

7  37 

2  59 

6400 

9 

18  6 

39 

5 

27 

11 

1500 

7  47 

2  35 

6500 

0  59 

18  28 

40 

6 

26 

10 

1600 

7  55 

2  14 

6600 

0  50 

18  48 

41 

6 

26 

9 

1700 

8  3 

1  53 

6700 

0  42 

19  6 

42 

7 

25 

8 

1800 

8  10 

1  35 

6800 

0  34 

19  21 

43 

7 

25 

7 

1900 

8  16 

1  18 

6900 

0  28 

19  33 

44 

7 

24 

6 

8000 

8  21 

1  3 

7000 

0  22 

19  44 

45 

8 

23 

5 

2100 

8  25 

0  51 

7100 

0  17 

19  52 

46 

8 

23 

4 

2200 

8  29 

0  40 

7200 

0  14 

19  57 

47 

9 

22 

3 

2300 

8  31 

0  32 

7300 

0  11 

20  0 

48 

9 

21 

2 

2400 

8  32 

0  25 

7400 

0  9 

20  1 

49 

10 

21 

1 

2500 

8  32 

0  21 

7500 

0  8 

19  59 

50 

10 

20 

0 

2600 

8  31 

0  19 

7600 

0  8 

19  55 

51 

10 

19 

99 

2700 

8  29 

0  20 

7700 

0  9 

19  48 

52 

11 

19 

98 

2800 

8  26 

0  23 

7800 

0  11 

19  40 

53 

11 

18 

»7 

2900 

8  23 

0  28 

7900 

0  15 

19  29 

54 

12 

17 

96 

3000 

8  18 

0  36 

8000 

0  19 

19  17 

55 

12 

17 

95 

3100 

8  12 

0  47 

8100 

0  24 

19  2 

56 

13 

16 

94 

3200 

8  6 

0^59 

8200 

0  30 

18  45 

!  57 

13 

15 

93 

3300 

7  58 

1  14 

8300 

0  37 

18  27 

58 

13 

15 

92 

3400 

7  50 

1  32 

8400 

0  45 

18  6 

59 

14 

14 

91 

3500 

7  41 

1  52 

8500 

0  53 

17  45 

60 

14 

14 

90 

3600 

7  31 

2  14 

8600 

3 

17  21 

61 

15 

13 

89 

3700 

7  21 

2  38 

8700 

13 

16  56 

62 

15 

13 

88 

3800 

7  9 

3  4 

8800 

25 

16  30 

63 

15 

12 

87 

3900 

6  58 

3  32 

8900 

36 

16  3 

64 

15 

12 

86 

4000 

6  45 

4  2 

9000 

49 

15  34 

65 

16 

11 

85 

4100 

6  32 

4  34 

9100 

2  2 

15  5 

66 

16 

11 

84 

4200 

6  19 

5  7 

9200 

2  16 

14  34 

67 

16 

11 

83 

4300 

6  5 

5  41 

9300 

2  30 

14  3 

68 

16 

10 

82 

4400 

5  51 

6  17 

9400 

2  45 

13  31 

69 

17 

10 

81 

4500 

5  36 

6  54 

9500 

3  0 

12  58 

70 

17 

10 

80 

4600 

5  21 

7  32 

9600 

3  16 

12  25 

71 

17 

10 

79 

4700 

5  6 

8  11 

9700 

3  32 

11  52 

72 

17 

10 

78 

4800 

4  51 

8  50 

9800 

3  48 

11  18 

73 

17 

10 

77 

4900 

4  35 

9  30 

9900 

4  4 

10  44 

74 

17 

9 

76 

,  5000 

4  20 

10  10 

10000 

4  20 

I  10  10 

75 

17 

9 

75 

TABLE   E. 


"      H 
§       K 


4 

I 


20 


TABLE    XVI. 
MOON'S  EPOCHS. 


Years 

1 

2 

3 

4 

5 

6 

7 

8 

• 

1846 

0013 

2475 

3275 

1688 

0773 

4880 

3179 

0800 

9542 

1847 

0006 

9683 

2941 

6432 

3245 

0678 

4239 

3257 

8406 

1848B. 

0026 

7542 

3646 

1463 

6052 

6847 

5358 

6106 

7295 

184) 

001  Jl 

4750 

3312 

6207 

8524 

2644 

6418 

8563 

6158 

1850 

0012 

1958 

2978 

0951 

0995 

8442 

7479 

1020 

5022 

1851 

0005 

9167 

2644 

5695 

3467 

4239 

8539 

3477 

3885 

1852B. 

0025 

7025 

3350 

0726 

6274 

0408 

9658 

6326 

2774 

1853 

0018 

4233 

3016 

5469 

8746 

6206 

0718 

8782 

1637 

1854 

0011 

1442 

2681 

0213 

1217 

2003 

1778 

1240 

0501 

1855 

0004 

8650 

2347 

4957 

3689 

7801 

2839 

3697 

9365 

1856  B. 

0024 

6509 

3053 

9988 

6496 

3970 

3957 

6446 

8254 

1857 

0017 

3717 

2719 

4732 

8968 

9767 

5018 

9002 

7117  1 

1858 

0010 

0925 

2385 

9476 

1439 

5565 

6078 

1460 

5981 

1859  » 

0003 

8134 

2051 

4220 

3911 

1362 

7139 

3917 

4845 

1860B. 

0023 

5992 

2756 

9551 

6718 

7531 

8257 

6765 

3734 

1861 
1862 

0016 
0009 

3200 
0409 

2423 

2088 

3995 
8739 

9190 
1661 

3329 
9126 

9317 
0378 

9222 
1679 

2597 
1461 

1863 

0002 

7617 

1754 

3483 

4133 

4923 

1438 

4137 

0324 

1864  B. 

0022 

5476 

2460 

8514 

6941 

1093 

2557 

6984 

9212 

1865 

0015 

2684 

2126 

3257 

9412 

6890 

3617 

9442 

8076 

1866 

0008 

9893 

1792 

8001 

1883 

26S7 

4678 

1899 

6940 

1867 

0001 

7101 

1457 

2745 

4355 

8485 

5738 

4357 

5804 

1868B. 

0021 

4959 

2163 

7776 

7163 

4654 

6857 

7204 

4692 

1869 

OC14 

2168 

1829 

2520 

9634 

0452 

7917 

9662 

3556 

1870 

0007 

9376 

1495 

7264 

2105 

6249 

8978 

2119 

2420 

1871 

0000 

6584 

1161 

2008 

4576 

2046 

0039 

4576 

1284 

1872  B. 

0020 

4432 

1867 

7039 

7383 

8215 

1158 

7423 

0172 

1873 

0013 

1640 

1533 

1783 

9854 

4012 

2239 

9880 

9036 

1874 

0006 

884S 

1199 

6527 

2325 

9809 

3300 

2337 

7900 

1875 

9999 

6056 

0865 

1271 

4796 

5606 

4361 

4794 

6764 

1876B. 

0019 

3914 

1571 

6292 

7603 

177o 

5480 

7641 

5652 

1877 

0012 

1122 

1247 

1036 

0074 

7572 

6541 

0098 

4516 

1878 

0005 

8330 

0913 

5780 

2545 

3369 

7602 

2555 

3380 

1879 

9998 

5538 

0579 

0524 

5016 

9166 

8663 

5012 

2244 

1880  B. 

0018 

3396 

1285 

5545 

7823 

5335 

9782 

7859 

1132 

1881 

0011 

0604 

0951 

0289 

0294 

1132 

0843 

0316 

9996 

1882 

0004 

7812 

0617 

5033 

2765 

6929 

1904 

2873 

8860 

1883 

9997 

5020 

0283 

9777 

5236 

2726 

2965 

5330 

7724 

1884  B. 

0017 

2878 

0989 

4798 

8043 

8895 

4084 

8177 

6612 

1885 

0010 

0086 

0655 

9542 

0514 

4692 

5145 

0634 

5476 

1886 

0003 

7294 

0321 

4286 

2985 

0489 

6206 

3091 

4340 

1887 

9996 

4502 

9987 

9030 

5456 

6286 

7267 

5548 

3204 

1888B. 

•  16 

2360 

0693 

4051 

8263 

2455 

8386 

8395 

2092 

1889 

0  09 

9568 

0359 

8795 

0734 

8252 

9447 

0852 

0956  . 

1890    0002 

6776 

0025 

3539 

3205  1  4049 

0508 

3309 

9820 

TABLE  XVI 
MOON'S  EPOCHS. 


21 


Years. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

1846 

203 

123 

250 

171 

419 

760 

126 

396 

167 

379 

204 

1847 

810 

484 

970 

644 

613 

9iJl 

486 

749 

643 

433 

371 

1848  B. 

486 

876 

759 

151 

905 

072 

881 

143 

144 

4r<7 

539 

1849 

093 

237 

479 

624 

099 

212 

241 

496 

619 

540 

705 

1850 

700 

597 

199 

097 

293 

352 

600 

848 

094 

594 

871 

1851 

306 

958 

918 

570 

487 

493 

960 

201 

569 

648 

038 

1852  B. 

983 

350 

707 

077 

780 

664 

355 

595 

070 

701 

206 

1853 

589 

711 

427 

550 

974 

804 

715 

948 

545 

755 

372 

1854 

196 

072 

147 

023 

168 

944 

074 

300 

020 

809 

539 

1855 

802 

432 

866 

496 

361 

085 

434 

653 

495 

863 

705 

1856  B. 

479 

824 

656 

003 

654 

256 

829 

047 

996 

916 

873 

1857 

086 

185 

375 

476 

848 

396 

189 

400 

471 

970 

039 

1858 

6'J2 

546 

095 

949 

042 

537 

548 

752 

947 

024 

206 

1859 

299 

907 

814 

422 

236 

677 

908 

105 

422 

078 

372 

1860B, 

975 

298 

604 

929 

529 

848 

303 

499 

923 

131 

540 

1861 

581 

659 

323 

402 

723 

988 

662 

852 

398 

185 

706 

1862 

187 

020 

042 

875 

916 

129 

021 

204 

873 

239 

873 

1863 

794 

381 

761 

348 

110 

269 

381 

557 

348 

292 

039 

1864  B. 

470 

773 

551 

855 

403 

440 

777 

951 

849 

346 

207 

1865 

077 

134 

271 

328 

597 

580 

136 

304 

324 

400 

373 

1866 

684 

494 

990 

801 

791 

721 

495 

657 

799 

453 

540 

1867 

290 

855 

710 

274 

985 

861 

855 

009 

274 

507 

707 

1868B, 

967 

247 

500 

781 

277 

032 

951 

404 

775 

561 

874 

1869 

573 

6D8 

219 

254 

471 

172 

610 

756 

251 

615 

040 

1870 

180 

96t 

938 

737 

665 

313 

969 

109 

726 

668 

207 

1871 

787 

328 

659 

200 

859 

554 

328 

562 

201 

721 

374 

1872  B. 

464 

720 

549 

707 

151 

725 

724 

957 

702 

785 

531 

1873 

071 

080 

269 

180 

345 

966 

083 

410 

177 

838 

698 

1874 

678 

440 

989 

653 

539 

205 

442 

863 

642 

891 

865 

1875 

285 

800 

709 

126 

733 

446 

801 

316 

117 

944 

032 

1876B. 

962 

192 

599 

633 

025 

617 

197 

711 

618 

008 

199 

1877 

569 

552 

319 

106 

219 

858 

556 

164 

093 

061 

366 

1878 

176 

912 

039 

579 

413 

099 

915 

617 

568 

114 

533 

1879 

783 

272 

759 

052 

607 

340 

274 

070 

043 

167 

700 

1880  B. 

460 

664 

649 

559 

899 

511 

670 

465 

544 

231 

867 

1881 

067 

024 

369 

032 

093 

752 

029 

918 

019 

284 

034 

1882 

674 

384 

089 

505 

287 

993 

388 

371 

494 

337 

201 

1883 

281 

744 

809 

978 

4S1 

234 

747 

824 

969 

390 

368 

1884  B. 

958 

136 

699 

485 

773 

405 

143 

219 

470 

454 

5H5 

1865 

565 

496 

419 

958 

967 

646 

502 

672 

945 

507 

702 

1886 

172 

856 

139 

431 

161 

887 

86i 

125 

420 

560 

869 

1887 

779 

216 

859 

904 

355 

128 

320 

578 

895 

613 

036 

1888  B. 

456 

608 

749 

411 

647 

299 

716 

973 

396 

677 

203 

1889 

063 

968 

469 

884 

841 

540 

075 

426 

871 

730 

370 

1890 

670 

328 

189, 

357 

035 

781 

434 

879 

346 

783 

537 

TABLE  XVII. 
MOON'S   MOTIONS   FOR   MONTHS. 


Months. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

i-   1  Com. 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

Jan'  J  Bis. 

9973 

9350 

8960 

9713 

9664 

9628 

9942 

9610 

9976 

rf  ,  1  Com. 
Feb'jBis. 

849 
851 

146 
9497 

2246 
12;»5 

88H6 
8609 

402 
66 

1533 
1161 

1789 
1731 

2099 
1709 

753 
729 

March  

1615 

8343 

1371 

6931 

9797 

1951 

3404 

3027 

1433 

April.. 

2464 

8490 

3616 

5827 

199 

3484 

5193 

5126 

2186 

May  

3285 

7986 

4J-22 

4436 

265 

4646 

6924 

680  5 

2914 

4134 

8133 

7067 

3332 

666 

6179 

8713 

8934 

S667 

July.  .  . 

4955 

7629 

8273 

1942 

732 

7341 

444 

643 

4:i96 

August...  . 

5804 

7776 

518 

838 

1134 

8874 

2233 

2742 

5148 

September  . 

6653 

7922 

2764 

9734 

1536 

408 

4021 

4842 

5901 

October  

7474 

7419 

3969 

8343 

1602 

1569 

5752 

655fr 

6630 

November.. 

8323 

7565 

6215 

7239 

2004 

3102 

7541 

8649 

7382 

December.  . 

9144 

7062 

7420 

•5848 

2070 

4264 

9272 

358 

8111 

TABLE  XVIII. 

MOON'S    MOTIONS   FOR  DAYS. 


Days. 

1 

2 

3 

4 

5 

6 

• 

7 

8 

9 

1 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

0000 

2 

27 

659 

1040 

287 

336 

372 

58 

390 

24 

3 

55 

1300 

2080 

574 

671 

744 

115 

781 

49 

4 

82 

1950 

3121 

861 

1007 

1116 

173 

1171 

73 

5 

109 

2600 

4161 

1148 

1342 

1488 

231 

1561 

97 

6 

137 

3249 

5291 

1435 

1678 

1860 

289 

1952 

121 

7 

164 

3899 

6241 

1722 

2013 

2232 

346 

2342 

146 

8 

192 

4549 

7281 

2009 

2349 

2604 

404 

2732 

170 

9 

219 

5199 

8321 

2236 

2684 

2976 

462 

3122 

194 

10 

246 

5849 

9362 

2583 

3020 

3348 

519 

3513 

219 

11 

274 

6499 

402 

2870 

3355 

3720 

577 

3903 

243 

12 

301 

7149 

~1442 

3157 

3691 

4093 

635 

4293 

267 

13 

328 

7799 

2482 

3444 

4026 

4465 

692 

4684 

291 

14 

356 

8449 

3522 

3731 

4362 

4837 

750 

5074 

316 

15 

383 

9098 

4563 

4018 

4698 

5209 

808 

5464 

340 

16 

411 

9748 

5603 

4305 

5033 

5581 

866 

5854 

364 

17 

438 

398 

6643 

4592 

5369 

5953 

923 

6245 

389 

18 

465 

1048 

7683 

4878 

5704 

6325 

981 

6635 

413 

19 

493 

1698 

8723 

5165 

6040 

6697 

1039 

7025 

437 

20 

520 

2348 

9763 

5452 

6375 

7069 

1096 

7416 

461 

21 

548 

2998 

804 

5739 

6711 

7441 

1154 

7806 

486 

22 

575 

3648 

1844 

6026 

7046 

7813 

1212 

8196 

510 

23 

602 

4298 

2884 

6313 

7382 

8185 

1269 

8586 

534 

24 

630 

4947 

3924 

6600 

7717 

8557 

1327 

8977 

559 

25 

657 

5597 

4964 

6887 

8053 

8929 

1385 

9367 

583 

26 

684 

6247 

6005 

7174 

8389 

9301 

1443 

9757 

607 

27 

712 

6897 

7045 

7461 

8724 

9673 

1500 

148 

631 

28 

739 

7547 

8085 

7748 

9060 

45 

1558 

538 

656 

29 

767 

8197 

9125 

8035 

9395 

417 

1616 

928 

680 

30 

794 

8847 

165 

8322 

9731 

789 

1673 

1319 

704 

31 

821 

9497 

1205 

8609 

66 

1161 

1731 

1709 

729 

TABLE  XVII. 

MOON'S   MOTIONS    FOB   MONTHS. 


Months. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

J-in  ]Com- 
Jtlil-  }  Hrs. 

pi  -  \  Com. 
1(>b-jBis. 
March  .  .  . 

000 
930 
175 
105 
IV) 

000 
969 
9H5 
934 
836 

000 
930 
184 
114 
157 

000 
966 
59 
25 
16 

000 
901 
74 
975 
F51 

000 
969 
946 
916 

8U1 

000 
963 
135 
98 
159 

000 
958 
304 
262 
482 

000 
974 

805 
779 
539 

000 
000 
5 
5 
9 

000 
000 
14 
14 
27 

April  
Miy  
June  
July.  .. 

H4 
419 
593 

6M8 

801 
735 
700 
634 

342 
556 
640 
754 

76 
101 
160 
185 

925 
899 
973 
948 

747 
663 
6)9 
525 

294 
392 
527 
625 

786 
47 
351 
613 

330 
11£ 

92) 
699 

13 

18 
22 
97 

41 
55 

*>9 
83 

August.  ..  . 

September  . 
October  
November.. 
December.  . 

873 

48 
152 
327 
432 

599 

563 

497 
462 
396 

938 

123 
237 
421 
535 

245 

304 

329 
388 
414 

22 

96 
71 
145 
120 

471 

417 
333 
279 
194 

759 

894 
992 
127 
225 

917 

221 

4t<3 

787 
49 

503 

30* 
81 
89* 
670 

31 

36 

£ 

4v 

97 

111 
125 
139 
353 

TABLE  XVIII. 

MOON'S    MOTIONS    FOR    DAYS. 


P,jys. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

%> 

1 

090 

000 

000 

090 

000 

000 

000 

090 

000 

000 

000 

2 

70 

31 

70 

34 

99 

31 

37 

42 

26 

0 

0 

3 

149 

62 

141 

68 

198 

61 

73 

84 

52 

0 

1 

4 

210 

93 

211 

103 

297 

92 

110 

126 

78 

0 

1 

5 

281 

125 

282 

137 

397 

122 

146 

168 

104 

1 

2 

6 

351 

156 

352 

171 

496 

153 

183 

210 

130 

1 

2 

7 

421 

187 

423 

205 

595 

183 

229 

252 

156 

1 

3 

8 

491 

218 

493 

239 

694 

214 

256 

294 

Ib2 

1 

3 

9 

561 

249 

564 

273 

793 

244 

233 

336 

2<)8 

1 

4 

10 

6-31 

28!) 

634 

398 

892 

275 

329 

379 

24 

1 

4 

11 

702 

311 

705 

342 

992 

305 

366 

421 

260 

1 

5 

12 

772 

342 

775 

376 

91 

336 

493 

463 

2-6 

2 

5 

13 

842 

374 

845 

410 

190 

366 

4^9 

505 

312 

2 

5 

14 

912 

495 

916 

444 

289 

397 

476 

547 

337 

2 

G 

15 

92 

436 

986 

478 

388 

427 

512 

589 

363 

2 

6 

16 

52 

467 

57 

513 

487 

458 

549 

631 

389 

2 

7 

17 

122 

498 

127 

547 

5S7 

488 

586 

673 

415 

2 

7 

18 

193 

529 

198 

581 

6*6 

519 

622 

715 

441 

2 

8 

19 

•263 

569 

263 

615 

785 

549 

659 

757 

467 

3 

8 

21 

333 

591 

339 

649 

884 

580 

695 

799 

493 

3 

9 

21 

4)3 

623 

409 

633 

983 

611 

722 

841 

517 

3 

9 

22 

473 

634 

480 

718 

82 

641 

769 

883 

545 

3 

10 

23 

543 

685 

550 

752 

182 

672 

805 

925 

571 

3 

10 

24 

614 

716 

621 

786 

281 

702 

842 

967 

597 

3 

11 

gr 

6S4 

747 

691 

829 

380 

733 

878 

9 

623 

4 

11 

26 

754 

778 

762 

854 

479 

7H 

915 

52 

649 

4 

11 

27 

824 

8!)9 

832 

888 

578 

794 

952 

94 

675 

4 

12 

28 

894 

840 

903 

923 

677 

824 

9h'8 

136 

701 

4 

12 

29 

964 

872 

973 

957 

777 

855 

25 

178 

707 

4 

13 

30 

34 

903 

43 

991 

876 

885 

61 

220 

753 

4 

13 

31 

105 

934 

114 

25 

975 

916 

98 

262 

779 

4 

14_J 

24 


TABLE   XIX. 
MOON'S   MOTIONS   FOB   HOURS. 


Hours.   1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

1 

27 

43 

12 

14 

16 

2 

16 

1 

2 

2 

54 

87 

24 

28 

31 

5 

33 

2 

3 

3 

81 

130 

36 

42 

47 

7 

49 

3 

4 

5 

108 

173 

48 

56 

62 

10 

65 

4 

5 

6 

135 

217 

60 

70 

78 

12 

81 

5 

6 

7 

162 

260 

72 

84 

93 

14 

98 

6 

7 

8 

190 

303 

84 

98 

109 

17 

114 

7 

8 

9 

217 

347 

96 

112 

124 

19 

130 

8 

9 

10 

244 

390 

JOS 

126 

140 

22 

146 

9 

10 

11 

271 

433 

120 

140 

155 

24 

163 

10 

11 

12 

298 

477 

131 

154 

171 

26 

179 

11 

12 

14 

325 

520 

143 

168 

186 

29 

195 

12 

13 

15 

352 

563 

155 

182 

202 

31 

211 

13 

14 

16 

379 

607 

167 

196 

217 

34 

228 

14 

15 

17 

406 

650 

179 

210 

233 

36 

244 

15 

16 

18 

433 

693 

191 

224 

248 

38 

260 

16 

17 

19 

460 

737 

203 

238 

264 

41 

276 

17 

18 

20 

487 

780 

215 

252 

279 

43 

293 

18 

19 

22 

515 

823 

227 

266 

295 

46 

309 

19 

20 

23 

542 

867 

239 

280 

310 

48 

325 

20 

21 

24 

569 

910 

251 

294 

326 

50 

341 

21 

22 

25 

596 

953 

263 

308 

341 

53 

358 

22 

23 

26 

623 

997 

275 

322 

357 

55 

374 

33 

24 

27 

650 

1040 

287 

336 

372 

58 

390 

24 

TABLE    XIX. 

MOON'S   MOTIONS   FOR   MINUTES. 


Min. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

1 

0 

0 

1 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

5 

0 

2 

4 

1 

1 

1 

0 

1 

0 

0 

0 

0 

0 

0 

10 

0 

5 

7 

2 

2 

3 

0 

3 

0 

0 

0 

0 

0 

1 

15 

0 

7 

11 

3 

3 

4 

1 

4 

0 

1 

0 

1 

0 

1 

20 

0 

9 

14 

4 

5 

5 

1 

5 

0 

1 

0 

1 

0 

1 

25 

0 

11 

18 

5 

6 

6 

1 

7 

0 

1 

1 

1 

2 

30 

14 

22 

6 

7 

8 

1 

8 

0 

1 

1 

3 

35 

16 

25 

7 

8 

9 

1 

10 

2 

2 

2 

40 

18 

29 

8 

9 

10 

2 

11 

2 

2 

3 

45 

20 

32 

9 

10 

12 

2 

12 

2 

2 

3 

50 

23 

36 

10 

11 

13 

2 

13 

2 

2 

3 

55 

25 

40 

11 

13 

14 

2 

15 

3 

3 

4 

CO 

27 

43 

12 

14 

15 

2 

16 

3 

3 

4 

TABLES. 


26 


HELIOCENTRIC    LONGITUDES,  ETC.  OF  THE  PLANET  TENUS,  AT  THE  TIMES  Of 

THE  NEXT  TWO  TKANSITS  OVER  THE  SUN'S  DISC. 
The  subject  matter  of  this  table  is  connected  with  Chapter  IX,  page  119. 


Times. 

Hel.  Long,  from 
true  Equinox. 

Hel.  Lat. 

Rad.  Vec. 

1874,  Dec.  8th,  at  12h. 
16h. 
20h. 

76°  41'    36.6" 
76    57    44.1 
77    13    51.5 

4'   6.3"  N. 
5    3.5 
6    1.0 

0.7203632 
0.7203449 
0.7203268 

1882,  Dec.  6th,  at  noon. 
4h. 
8h. 

74    12    55.7 
74    29      2.5 
74    45      9.7 

4  58.1  S. 
4    0.8 
3    3.5 

0.7205502 
0.7205315 
0.7205127 

DIP   OF   THE   HORIZON. 
For  the  principle  of  computing  the  dip  of  the  horizon  see  text-note,  page  54. 


Might 
fe'et. 

Dip. 

T 

feet. 

Dip. 

1 

1'   1" 

16 

4'   3" 

2 

1   26 

17 

4   11 

3 

1   45 

18 

4   18 

4 

2   2 

19 

4   25 

5 

2   16 

20 

4   32 

6 

2   29 

21 

4   39 

7 

2   41 

22 

4   45 

8 

2   52 

23 

4   52 

9 

3   2 

24 

4   58 

10 

3   12 

25 

5   4 

11 

3   22 

26 

5   10 

12 

3   31 

28 

5  22 

13 

3   39 

30 

5   33 

14 

3   48 

35 

6   1 

15 

3   56 

40 

6  25 

«TTN'S  SEMIDIAMETER  FOR  EVERY  TENTH  DAT  OF  THE  TEAK 


Days. 

1 
11 
21 

Jan. 
/   n 
16  18 
16  17 
16  17 

July. 

15  46 
15  46 
15  46 

Days. 

1 
11 
21 

April. 
/   // 
16   1 
15  58 
15  55 

Oct. 
/   // 
16   1 
16   3 
16   7 

1 
11 
21 

Feb. 
16  15 
16  13 
16  11 

August. 
15  47 
15  49 
15  51 

1 
11 

21 

May. 
15  53 
15  51 

15  49 

Nor. 
16   9 
16  12 
16  14 

1 
11 
21 

March. 
16  10 
16   7 
16   4 

Sept 
15  53 
15  56 

15  58 

1 
11 
21 

June. 
15  48 
15  46 
15  46 

Dec. 
16  16 
16  17 

16  18 

26 


TABLE  XX. 

MOON'S  EPOCHS. 


Yeure. 

Erection. 

Anomaly. 

Variation. 

Longitude. 

1846 
1847 

1848  B. 
1849 
1850 

s   o   '   « 
2   0  45   6 
7  21  16  35 
1  23   7   5 
7  13  38  35 
1   4  10  4 

s   °   '   " 
0  26  21   2 
3  25  4  23 
7   6  51  37 
10  5  34  57 
1   4  18  18 

s   o      " 
1   5  48   4 
5  15  25  29 
10  7  14  21 
2  16  51  46 
6  26  29  11 

8   0    '    " 

10  15  48  23 
2  25  11  28 
7  17  45  8 
11  27   8  14 
4  6  31  20 

1851 
1852  B. 
1853 
1854 

1855 

6  24  41  35 
0  26  32  5 
6  17   3  34 
0  7  35  4 
5  23  6  33 

4  3  1  38 
7  14  48  53 
19  13  32  13 
1  12  15  34 

4  10  58  54 

11   6  6  36 
3  27  55  29 
8  7  32  53 
0  17  10  19 
4  26  47  43 

8  15  54  25 
1   8  23  6 
5  17  51  11 
9  27  14  17 
2   6  37  22 

1856  B. 
1957 
1858 
1859 
1860B. 

11  29  57   3 
5  20  28  33 
11  11   02 
5  1  31  33 
11   3  22  3 

7  22  46  9 
10  21  29  29 
1  20  12  50 
4  18  56  10 
8  0  43  25 

9  18  36  36 
1  28  14   1 
6  7  51  26 
10  17  28  52 
3  9  17  44 

6  29  11   3 
11   8  34  9 
3  17  57  14 
7  27  20  20 
0  19  54  0 

1861 
1862 
1863 
1864  B. 
1865 

4  23  53  33 
10  14  25  3 
4  4  56  33 
10  6  47  2 
3  27  18  32 

10  29  26  45 
1  28  10  6 
4  26  53  27 
8  8  43  41 
11   7  24  2 

7  18  55   9 
11  28  32  34 
4  8  10  0 
8  29  58  51 
1   9  36  17 

4  29  17   6 
9  8  40  12 
1  18  3  18 
6  10  36  58 
10  20  0  4 

1866 
1867 
1868  B. 
1869 
1870 

9  17  59  2 
3  8  21  32 
9  10  J2  2 
3  0  43  33 
8  21  15  2 

2  6  7  23 
5  4  50  43 
8  16  37  58 
11  15  21  19 
2  14  4  43 

5  19  13  42 
9  28  51   8 
2  20  40  0 
7   0  17  25 
11   9  54  50 

2  29  23  10 
7  8  46  15 
0  1  19  56 
4  10  43  2 
8  20  6  8 

1871 
1872  B. 
1873 
1874 
1875 

2  11  45  31 
8  2  17   0 
2  4  7,31 
7  24  39   0 
1  15  10  29 

5  12  47   1 
8  11  30  21.7 
11  23  17  36.G 
2  22  0  57.3 
5  23  44  18 

3  19  31  1C 
7  29  8  41 
0  23  57  36 
5  0  35  0 
9  10  12  24 

0  29  28  13.7 
5  8  51  19.4 
10  1  25  0.3 
2  10  48   6 
6  20  11  11.7 

1876  B. 

1877 
1878 
1879 
1880B. 

7  5  41  59 
1   7  32  30 
6  28  3  59 
0  18  35  28 
6  9  6  58 

8  19  27  38.7 
0  1  14  53.6 
2  29  58  14.3 
5  28  41  35 
8  27  24  55.7 

1  19  49  50 
6  11  38  40 
10  21  16  5 
3  0  53  30 
7  10  30  55 

10  29  34  17.4 
3  22  7  58.3 
8  1  31   4 
0  10  54  9.7 
4  20  17  15.4 

1881 

1882 
1883 
1884  B. 
1885 

0  10  57  29 
6   1  28  58 
11  22  0  27 
5  12  31  56 
11  14  22  28 

0  9  12  10.6 
3  7  55  31.3 
6  6  38  52.0 
9  5  22  12.7 
0  17  9  27.6 

0  2  19  47 
4  11  57  12 
8  21  34  37 
1   1  12  2 
5  23  0  54 

9  12  50  56.3 
1  22  14  2.0 
6  1  37  7.7 
10  11   0  13.4 
3  3  33  54.3 

1886 
1887 
1888B. 
J889 
1890 

5  4  53  57 
10  25  25  26 
4  15  56  57 
10  17  47  28 
4  8  18  57 

3  15  52  48.3 
6  14  36  9.0 
9  13  19  29.7 
0  25  6  44.6 
3  23  50  5.3 

10  2  38  19 
2  12  15  44 
6  21  53   9 
11  13  42   1 
3  23  19  26 

7  12  57  0.0 
11  22  20  5.7 
4  1  43  11.0 
8  24  15  51.9 
1   3  39  57.6 

TABLE    X      . 
MOON'S  EPOCHS. 


Years. 

Supp.  of  Node. 

II. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

1846 

4  16  :5   9 

s   °   ' 
11   7  56 

254 

258 

937 

941 

847 

113 

1847 

5  5  54  52 

2  28  38 

668 

670 

245 

247 

927 

053 

1848  B 

2  25  17  45 

709 

116 

122 

5b2 

587 

042 

997 

1849 

6  14  37  27 

10  20  41 

531 

535 

889 

893 

122 

937 

1850 

7  3  57   9 

2  11  13 

944 

947 

196 

200 

202 

876 

1851 

7  23  16  51 

6  1  45 

358 

359 

504 

506 

282 

816 

1852  B. 

8  12  39  44 

10  3  27 

806 

811 

841 

846 

398 

760 

1853 

9  1  59  26 

1  23  59 

220 

223 

148 

152 

477 

700 

1854 

9  21  19  9 

5  14  31 

6?4 

6.J6 

456 

459 

557 

639 

1855 

10  10  38  51 

953 

047 

048 

76J 

765 

637 

579 

1856  B. 

11   0   1  44 

1   6  44 

495 

500 

100 

105 

753 

523 

1857 

11  19  21  26 

4  27  16 

909 

912 

407 

411 

832 

463 

1858 

0  8  41   8 

8  17  48 

323 

S25 

715 

718 

912 

402 

1859 

0  28  0  51 

0  8  20 

7S6 

737 

023 

024 

992 

342 

1860  B, 

1  17  23  43 

4  10   1 

184 

189 

359 

364 

108 

2-6 

1861 

2  6  43  27 

8  0  33 

598 

601 

666 

670 

187 

226 

1862 

2  26  3  9 

11  21   5 

012 

014 

974 

977 

267 

165 

1863 

3  15  23  11 

3  11  37 

426 

426 

2b2 

283 

347 

105 

1864  B. 

4  4  45  44 

7  13  18 

873 

878 

618 

623 

463 

049 

1865 

4  24  5  46 

11   3  50 

287 

291 

926 

929 

542 

989 

1866 

5  13  25  28 

2  24  22 

701 

703 

233 

236 

622 

928 

1867 

6  2  45  10 

6  14  54 

115 

115 

544 

542 

702 

868 

;  ll>68  B. 

6  22  7  43 

10  16  36 

563 

567 

877 

882 

818 

812 

!  1869 

7  11  27  46 

278 

977 

980 

185 

188 

897 

752 

1870 

8  0  47  28 

5  27  40 

390 

392 

493 

495 

977 

691 

1871 

8  20  6  49 

9  18  11 

803 

804 

800 

802 

057  630 

1872  B. 

9   9  26  31 

1   8  43 

216 

216 

108 

110 

137 

569 

1873 

9  28  49  24 

5  10  25 

664 

668 

444 

450 

252 

514 

1874 

10  18  9   6 

9   0  57 

077 

080 

752 

758 

332 

453 

1875 

11   7  28  48 

0  21  29 

490 

492 

054 

064 

412 

392 

1876  B. 

11  26  43  31 

4  12   1 

904 

905 

364 

370 

492 

331 

1877 

0  16  11  24 

8  13  42 

352 

357 

700 

710 

607 

276 

1878 

1   5  31   6 

0  4  14 

765 

769 

008 

018 

687 

215 

1879 

1  24  50  48 

3  24  46 

178 

181 

316 

326 

767 

154 

1880  B. 

2  14  10  30 

7  15  18 

593 

593 

624 

630 

847 

093 

1881 

3   3  33  23 

11  16  59 

041 

045 

960 

970 

962 

038 

1882 

3  22  53  5 

3  7  31 

454 

457 

268 

278 

042 

977 

1883 

4  12  12  47 

6  28   3 

867 

869 

576 

5F6 

122 

916 

1884  B. 

5   1  32  29 

10  18  35 

280 

281 

884 

894 

202 

855 

1885 

5  20  55  22 

2  20  16 

728 

733 

220 

234 

317 

800 

1886 

6  10  15  4 

6  10  48 

141 

145 

52S 

542 

397 

730 

1887 

6  29  34  46 

10  1  20 

554 

557 

836 

850 

477 

678 

1888  B. 

7  18  54  28 

1  21  52 

967 

969 

144 

158 

557 

617 

1889 

8  8  17  21 

5  23  33 

415 

421 

480 

498 

672 

562 

1890 

8  27  36  3 

9  14   5 

828 

833  788 

806 

752 

501 

TABLE    XX. 

MOON'S   MOTIONS    FOR    MONTHS. 


Months. 

Evection. 

Anomaly. 

Variation. 

M.  Longitude. 

*»-l£r: 

HmT. 

March      .... 

6        0         '          " 

0000 
11    18    41      1 
11    20    48   42 
11      9    29    43 
10      7    40    26 

8         0          '          » 

0000 
11    16    56      6 
1    15     0    53 
1      1    56    59 
1    20    50     4 

8        0 

0000 
11    17    48    33 
0    17    54    48 
0     5    43    21 
11    29    15    15 

8        0                   » 

0000 
11    16   49    25 
1    18    28     6 
1      5    17    31 
1    27    24    27 

April 

9    28    29      8 

3      5    50    57 

0    17    10      3 

3    15    52    32 

May  

9      7    58    51 

4      7    47    56 

0    22    53    24 

4    21    10     3 

8    28    47    33 

5    22    48    49 

1    10   48    11 

6      9    38      9 

July.. 

8      8    17    16 

6    24    45    48 

1    16    31    32 

7    14    55    40 

August  

September..  . 
October  

7    29      5    59 

7    19    54   41 
6    29    24   24 

8      9    46   42 

9    24   47    35 
10    26    44    34 

2     4   26    20 

2    22    21      7 

2   28     4   28 

9      3    23    46 

10   21    51    52 
11    27      9    22 

November.  .  . 
De6ember  .  .  . 

6    20    13      6 
5    29    42   49 

0    11    45    27 
1    13    42    26 

3    15    59    16 
3    21    42    37 

1    15    37    28 
2    20    54    59 

T 

MOON 

ABLE    X: 

's   MOTIONS   F01 

£. 

El   DAYS. 

Days. 

Evection. 

Anomaly. 

Variation. 

Mean  Longitude. 

1 

Os     1°    0'    0" 

Os     GO     0'      0" 

Os     GO    0'      0" 

Os     GO    0'     0" 

2 

0    11     18   59 

0    13      3    54 

0    12    11     27 

0    13    10    35 

3 

0    22    37    59 

0    26      7    48 

0    24    22    53 

0    26    21     10 

4 

1      3    56   58 

1      9     11     42 

1       6    34    20 

1       9    31     45 

5 

1     15    15    58 

1    22    15    36 

1     18    45    47 

1     22    42    20 

6 

1    26    34    57 

2      5     19    30 

2      0    57     13 

2      5    52    55 

7 

2      7    53    57 

2    18    23    24 

2    13      8    40 

2    19      3    30 

8 

2    19    12    56 

3      I     27     18 

2    25    20      7 

3      2    14      5 

9 

3      0    31    55 

3    14    31     12 

3      7    31     34 

3    15    24    40 

10- 

3    11    50    55 

3    27     35      6 

3    19    43      0 

3    28    35    15 

11 

3    23      9    54 

4    10    39      0 

4      1     54    27 

4    11     45    50 

12 

4      4    28   54 

4    23    42    54 

4    14      5    54 

4    24    56    25 

13 

4    15    47    53 

5      6    46    48 

4    26    17    20 

5870 

14 

4    27      6    53 

5    19    50    42 

5      8    28    47 

5    21     17    35 

15 

5      8    25    52 

6      2    54    36 

5    20    40    14 

6      4    28    10 

16 

5    19    44   51 

6    15    58    29 

6      2    51     40 

6    17    38    45 

17 

6      1      3    51 

6    29      *>    23 

6    15      3      7 

7      0    49    20 

18 

6    12    22   50 

7    12      6    17 

6    27     14    34 

7     13    59    55 

19 

6    23    41    50 

7    25     10    11 

7      9    26      1 

7    27     10    30 

20 

7      5      0    49 

8      8     14      5 

7    21     37    27 

8    10    21      5 

21 

7    16    19    49 

8    21     17    59 

8      3    48    54 

8    23    31    40 

22 

7    27    38    48 

9      4    21    53 

8    16      0    21 

9      6    42    16 

23 

8      8    57    47 

9    17    25    47 

8    28     11     47 

9    19    52    51 

24 

8    20    16    47 

10      0    29    41 

9    10    23    14 

10      3      3    26 

25 

9      1     35    46 

10    13    33    35 

9    22    34    41 

10    16    14      1 

26 

9    12    54   46 

10    26    37    29 

10      4    46      7 

10    29    24    36 

27 

9    24    13    45 

11      9     41    23 

10    16    57    34 

11     12    35    11 

28 

10      5    32    45 

11    22    45    17 

10    29      9      1 

11     25    45    46 

29 

10    16    51    44 

0      5     49    11 

11     11     20    28 

0      8    56    21 

30 

10    28    10    43 

0    18    53      5 

11    23    31     54 

0    22      6    56 

31 

11      9    29    43 

1      1     56    59 

0      5    43    21 

1      5    17    31   / 

TABLE    XX. 
MOON'S   MOTIONS   FOR   MONTHS. 


Months. 

Supp.  of  Node. 

II. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

T  I  Com.. 
Jan'jBis.  . 
,-,,"1  Com.. 
Feb'jBis.  . 
March  

8         0          '          " 

0000 
11     29    56    49 
0      1     38    30 
0      1     35    19 
0      3      7    27 

8         0          ' 

000 
11     18    51 
11     15    43 
11      4    34 
9    27    59 

000 
966 
54 

20 
7 

000 
961 
224 

185 
330 

000 
972 

875 
847 
666 

000 
966 
45 
11 
989 

000 
964 
111 
75 
114 

000 
995 
165 
159 
313 

April  

0      4    45    57 

9     13    42 

61 

554 

542 

34 

995 

478 

May  

0      6    21     16 

8    18    15 

81 

738 

389 

46 

300 

638 

J  UI16  ...... 

0      7    59    46 

8      3    58 

136 

969 

964 

91 

411 

802 

July 

0      9    35      5 

7      8    32 

1^6 

147 

119 

103 

486 

962 

August  

September..  . 
October  
November.  .. 
December  .  .  . 

0    11     13    35 

0    12    52      5 
0    14    27    24 
0    16      5    53 
0    17    41     13 

6    24    15 

6      9    58 
5    14    32 
5      0     15 
4      4    49 

210 

265 
285 
339 
359 

371 

595 

780 
4 

188 

987 

862 
710 

585 
432 

147 

193 

204 
250 
261 

497 

708 
783 
894 
969 

126 

291 
451 
615 

775 

TABLE    XX. 

MOON'S    MOTIONS    FOR   DAYS. 


Days 

Supp.  of  Node 

II. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

1 

00  0'  0" 

Os  GO  0' 

000 

000 

000 

000 

000 

000 

2 

0   3  11 

11   9 

34 

39 

28 

34 

36 

5 

3 

0   6  21 

22  18 

68 

79 

56 

67 

72 

11 

4 

0   9  32 

1   3  27 

102 

118 

85 

101 

108 

16 

5 

0  12  52 

1  14  37 

136 

158 

113 

135 

143 

21 

6 

0  15  53 

1  25  46 

170 

197 

141 

169 

179 

27 

8 

0  19   4 

2   6  55 

204 

237 

169 

202 

215 

32 

8 

0  22  14 

2  18   4 

238 

276 

198 

236 

251 

37 

9 

0  25  25 

2  29  13 

272 

316 

226 

270 

287 

43 

10 

0  28  36 

3  10  22 

306 

355 

254 

303 

323 

48 

11 

0  31  46 

3  21  31 

340 

395 

282 

337 

358 

53 

12 

0  34  57 

4   2  40 

374 

434 

311 

371 

394 

58 

13 

0  38   7 

4  13  50 

408 

474 

339 

405 

430 

64 

14 

0  41  18 

4  24  59 

442 

513 

367 

438 

466 

69 

15 

0  44  29 

568 

476 

553 

395 

472 

502 

74 

16 

0  47  39 

5  17  17 

510 

592 

424 

506 

538 

80 

17 

0  50  50 

5  28  26 

544 

632 

452 

539 

573 

85 

18 

0  54   1 

6   9  35 

578 

671 

480 

573 

609 

90 

19 

0  57  11 

6  20  44 

612 

711 

508 

607 

645 

96 

20 

0  22 

7   1  53 

646 

750 

537 

641 

681 

101 

21 

3  33 

7  13   3 

680 

790 

565 

674 

717 

1% 

22 

6  43 

7  24  12 

714 

829 

593 

708 

753 

112 

23 

9  54 

8   5  21 

748 

869 

621 

742 

788 

117 

24 

13   5 

8  16  30 

782 

908 

650 

775 

824 

122 

25 

16  15 

8  27  39 

816 

948 

678 

809 

860 

128 

26 

19  26 

9   8  48 

850 

987 

706 

843 

896 

133 

27 

22  36 

9  19  57 

884 

027 

734 

877 

932 

138 

28 

25  47 

10   1   6 

918 

066 

762 

910 

968 

143 

29 

28  58 

10  12  16 

952 

106 

791 

944 

003 

149 

30 

32   8 

10  23  25 

986 

145 

819 

978 

039 

154 

31 

35  19 

11   4  34 

020 

185 

847 

Oil 

075 

151 

30 


TABLE    XX. 
MOON'S   MOTIONS    FOR   HOURS. 


Hours. 

Evection. 

Anomaly. 

Variation 

Longitude. 

1 
2 
3 
4 
5 

0    '    " 

0  28  17 
0  56  35 
1  24  52 
1  53  10 
2  21  27 

O    '    " 

0  32  40 
1   5  19 
1  37  59 
2  10  39 
2  43  19 

O    '    a 

0  30  29 
1   0  57 
1  31  26 
2   1  54 
2  32  23 

O    '    " 

0  32  56 
1   5  53 
1  38  49 
2  11  46 
2  44  42 

6 
7 
8 
9 
10 

2  49  45 
3  18   2 
3  46  20 
4  14  37 
4  42  55 

3  15  58 
3  48  38 
4  21  18 
4  53  58 
5  26  37 

3   2  52 
3  33  20 
4   3  49 
4  34  17 
5   4  46 

3  17  39 
3  50  35 
4  23  32 
4  56  28 
5  29  25 

11 
'12 
13 
14 
15 

5  11  12 
5  39  30 
6   7  47 
6  36   5 
7   4  22 

5  59  17 
6  31  57 
7   4  37 
7  37  16 
8   9  56 

5  35  15 
6   5  43 
6  36  12 
7   6  40 
7  37   9 

6   2  21 
6  35  17 
7   8  14 

7  41  10 
8  14   7 

16 
17 

18 
19 
20 

7  32  40 
8   0  57 
8  29  15 
8  57  32 
9  25  50 

9  42  36 
8  15  16 
9  47  55 
10  20  35 
10  53  15 

8   7  38 
8  38   6 
9   8  35 
9  39   3 
10   9  32 

8  47   3 
9  20   0 
9  52  56 
10  25  53 

10  58  49 

21 
22 
23 
24 

9  54   7 
10  22  24 
10  50  42 
11  18  59 

11  25  55 

11  58  34 
12  31  14 
13   3  54 

10  40   1 
11  10  29 
11  40  58 
12  11  27 

11  31  46 
12   4  42 
12  37  39 
13  10  35 

TABLE    XXI. 

MOON'S    MOTIONS    FOR    MINUTES. 


Min. 

Evec. 

Anomaly. 

Variations. 

Longitude. 

Sup. 
Node. 

k  II. 

1 

0  28 

0  33 

0  30 

0  33 

0 

0 

5 

2  21 

2  43 

2  32 

2  45 

1 

2 

10 

4  43 

5  27 

5   5 

5  29 

1 

5 

15 

7   4 

8  10 

7  37 

8  14 

2 

7 

20 

9  26 

10  53 

10  10 

10  59 

3 

9 

25 

11  47 

13  37 

12  42 

13  43 

3 

12 

30 

14   9 

16  20 

15  14 

16  28 

4 

14 

35 

16  30 

19   3 

17  47 

19  13 

5 

16 

40 

18  52 

21  46 

20  19 

21  58 

5 

19 

45 

21  13 

24  30 

22  52 

24  42 

6 

21 

50 

23  34 

27  13 

25  24 

27  27 

7 

23 

55 

25  56 

29  56 

27  56 

30  12 

7 

26 

(VI 

28  17 

32  4!) 

30  29 

32  56 

8 

28 

TABLE    XX. 

MOON'S   MOTIONS   FOR    HOURS. 


31 


Hours. 

j  _:  

iSupn.  of 

Node. 

II. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

/   / 

0   ' 

1 

0   8 

0  28 

1 

2 

1 

1 

1 

0 

2 

0  1G 

Of\  A 

0  56 

If)  A 

3 

3 

2 

3 

3 

0 

4 

mm 

0  32 

m 
I  52 

6 

7 

5 

6 

6 

1 

5 

0  40 

2  19 

7 

» 

6 

7 

7 

1 

6 

0  48 

2  47 

9 

n 

7 

9 

9 

1 

7 

0  56 

3  15 

10 

13 

8 

10 

10 

2 

8 

4 

3  43 

11 

13 

9 

11 

12 

2 

9 

11 

4  11 

13 

15 

11 

13 

13 

2 

10 

19 

4  39 

14 

16 

12 

14 

15 

2 

31 

27 

5   7 

16 

18 

13 

15 

16 

2 

12 

35 

5  35 

17 

20 

14 

17 

18 

3 

13 

43 

6   2 

18 

21 

15 

18 

19 

3 

14 

51 

6  30 

20 

23 

16 

19 

21 

3 

15 

59 

6  58 

21 

25 

18 

21 

22 

3 

16 

2   7 

7  26 

23 

26 

19 

22 

24 

4 

17 

2  15 

7  54 

24 

28 

20 

24 

25 

4 

18 

2  23 

ft  22 

26 

29 

21 

25 

27 

4 

19 

2  31 

8  50 

27 

31 

22 

27 

28 

4 

20 

2  39 

9  18 

28 

32 

24 

28 

30 

4 

21 

2  47 

9  45 

30 

34 

25 

29 

31 

5 

22 

2  55 

10  13 

31 

36 

26 

31 

33 

5 

23 

3   3 

10  41 

33 

38 

27 

32 

34 

5 

24 

3  11 

11   9 

34 

39 

28 

34 

36 

5 

TABLE    A.* 

PERTURBATIONS    OF    EARTH'S 
RADIUS    VECTOR. 


TABLE    B. 

APPROX.  LAT. ARG.  N. 


Arg. 

I. 

II. 

III. 

Arg. 

I. 

II. 

III. 

0 

8 

4 

3 

500 

2 

0 

4 

50 

8 

4 

3 

550 

2 

1 

4 

100 

7 

4 

2 

600 

3 

1 

3 

150 

7 

4 

1 

650 

3 

2 

2 

200 

6 

4 

0 

700 

4 

3 

1 

250 

5 

4 

0 

750 

5 

4 

0 

300 

4 

3 

1 

800 

6 

4 

0 

350 

3 

2 

2 

850 

7 

4 

1 

400 

3 

1 

3 

900 

7 

4 

2 

450 

2 

1 

4 

950 

8 

4 

3 

500 

2 

0 

4 

1000 

8 

4 

3 

N. 
A. 

N. 
D. 

S. 
D. 

S. 
A. 

f>'s 

Lat. 

0 

500 

500 

1000 

t   a 

0   0 

5 

495 

505 

995 

9  41 

10 

490 

510 

990 

19  22 

15 

485 

515 

985 

29   3 

20 

480 

520 

980 

38  40 

25 

475 

525 

975 

48  18 

30 

470 

530 

970 

58  40 

35 

465 

535 

965 

67  28 

40 

460 

540 

960 

76  45 

45 

455 

545 

955 

86  21 

50 

450 

550 

950 

95  26 

55 

445 

555 

945 

04  56 

•  Tables  A.  and  B.  are  put  in  this  place  on  account  of  the  convenience  in  the  page. 


32  TABLE    XXI 

fIRST  EQUATION  OF  MOON's  LONGITUDE. — ARGUMENT  1. 


Arg. 

1 

Diff. 

Anr 

1 

Diff. 

0 
100 

12  40 
11   58 

42 

5000 
5100 

it 

12  40 
13  20 

40 

200 

11   16 

42 

*c\  . 

5200 

14   1 

41 

300 

10  34 

•I 

A  1 

5300 

14  41 

40 

400 

9  53 

41 

5400 

15  20 

39 

500 

9  12 

41 

Ai\ 

5500 

16   0 

40 

600 

8  32 

40 

qQ 

5600 

16  38 

38 

q7 

700 

7  54 

oo 

qo 

5700 

17  15 

O* 

Q7 

800 

7  16 

OU 

Qfi 

5800 

17  52 

o/ 

OK 

900 

6  40 

oO 

5900 

18  27 

OO 

1000 

6   6 

no 

6000 

19   1 

34 

1100 

5  33 

33 

6100 

19  33 

32 
qi 

1200 

5   2 

OA 

6200 

20   4 

ol 

cin 

1300 

4  32 

oU 

Cfc*7 

6300 

20  33 

sv 

CfcO 

1400 
1500 

4   5 

3  40 

mi 

25 

qq 

6400 
6500 

21   1 
21   27 

28 
26 

1600 

3  17 

20 

91 

6^00 

21   50 

oo 

1700 

2  56 

Zl 

IP. 

6700 

22  12 

SESf 
-i(i 

1800 

2  38 

lo 

6800 

22  31 

IJ 

1900 

2  22 

1  O 

6900 

22  48 

2009 

2   9 

lo 

7000 

23   3 

15 

t  Ck 

2100 

58 

7100 

23  15 

IB 

in 

2200 

50 

7200 

23  25 

ill 

2300 

44 

'  -   ^ 

7300 

23  32 

2400 

41 

7400 

23  37 

2500 

41 

7500 

23  39 

2600 

43 

c 

7600 

23  39 

lag 

2700 

48 

O 

7700 

23  36 

2800 

55 

-I  f\ 

7800 

23  30 

2900 
3000 

2   5 
2  -17 

10 
12 

7900 
8000 

23  22 
23  11 

11 

1  Q 

3100 

2  32 

17 

8100 

22  58 

Jo 
1  u 

3200 
3300 

2  49 
3   8 

1  1 
19 

8200 
8300 

22  42 
22  24 

ID 

18 

Cll 

3400 

3  30 

00 

8400 

22   3 

21 

99. 

3500 
3600 

3  53 
4  19 

26 

97 

8500 
8600 

21  40 
21  15 

ZJ 

25 

97 

3700 
3800 

4  46 
5  16 

ml 

30 

Ol 

8700 
8800 

20  48 
20  18 

ml 

30 

Ol 

3900 
4000 
4100 
4200 

5  47 
6  19 
6  53 

7  28 

ol 

32 
34 
35 

q7 

8900 
9000 
9100 
9200 

19  47 
19  14 
18  40 

18   4 

31 
33 
34 
36 

qo 

4300 

8   5 

Ol 
07 

9300 

17  26 

oO 
qo 

4400 

8  42 

ol 

qo 

9400 

16  48 

oO 

An 

4500 

9  20 

OO 
Oil 

9500 

16   8 

4U 

At 

4600 

9  59 

39 

A(\ 

9600 

15  27 

41 

At 

4700 

10  39 

4U 

9700 

14  46 

41 
An 

4600 

11  19 

JJ 

9800 

14   4 

Vm 

AC) 

4909 

11  59 

J.1 

9900 

13  22 

VI 

5000 

12  40 

41 

10000 

12  40 

TABLE    XXII. 

EQUATIONS  2  TO  7  OF  MOON'S  LONGITUDE. ARGUMENTS  2  TO  7. 


33 


Arg. 

2 

3 

4 

5 

6 

7 

Arg. 

2500 

4  57 

0   2 

6  30 

3  39 

0   6 

0   1 

2500 

2600 

4  57 

0   2 

6  30 

3  39 

0   6 

0   1 

2400 

2700 

4  56 

0   3 

6  29 

3  38 

0   7 

0   1 

2300 

2800 

4  55 

0   3 

6  27 

3  37 

0   8 

0   2 

2200 

2900 

4  53 

0   4 

6  24 

3  36 

0   9 

0   3 

2100 

3000 

4  50 

0   5 

6  21 

3  34 

0  10 

0   4 

2000 

3100 

4  47 

0   6 

6  17 

3  32 

0  12 

0   5 

1900 

3200 

4  43 

0   8 

6  12 

3  29 

0  14 

0   6 

1800 

3300 

4  39 

0   9 

6   7 

3  26 

0  17 

0   8 

1700 

3400 

4  34 

0  11 

6   1 

3  22 

0  19 

0  10 

1600 

3500 

4  29 

0  13 

5  54 

3  18 

0  22 

0  12 

1500 

3600 

4  23 

0  15 

5  47 

3  14 

0  23 

0  14 

1400 

3700 

4  17 

0  18 

5  39 

3  10 

0  29 

0  17 

1300 

3800 

4  11 

0  20 

5  30 

3   5 

0  33 

0  19 

1200 

3900 

4   4 

0  23 

5  21 

3   0 

0  37 

0  22 

1100 

400(1 

3  57 

0  26 

5  12 

2  54 

0  41 

0  25 

1000 

4100 

3  49 

0  29 

5   2 

2  49 

0  45 

0  28 

900 

4200 

3  41 

0  32 

4  52 

2  43 

0  50 

0  31 

800 

4300 

3  33 

0  35 

4  41 

2  37 

0  54 

0  35 

700 

4400 

3  24 

0  39 

4  30 

2  30 

0  59 

0  38 

600 

4500 

'A  15 

0  42 

4  19 

2  24 

4 

0  42 

500 

4600 

3   7 

0  46 

4   7 

2  17 

9 

0  45 

400 

4700 

2  58 

0  49 

3  56 

2  10 

14 

0  49 

300 

4-00 

2  48 

0  53 

3  44 

2   4 

19 

0  53 

200 

4930 

2  39 

0  56 

3  32 

1  57 

25 

0  56 

100 

5000 

2  30 

0 

3  20 

1  50 

30 

1   0 

0000 

510) 

2  21 

4 

3   8 

1  43 

35 

1   4 

9900 

5200 

2  11 

7 

2  56 

1  36 

40 

1   7 

9800 

5300 

2   2 

11 

2  44 

1  29 

46 

1  11 

9700 

5400 

1  53 

14 

2  33 

1  23 

1  51 

1  15 

9600 

S.iOO 

1  44 

18 

2  21 

1  16 

1  56 

1  18 

9500 

5600 

1  36 

21 

2  10 

1  10 

2   1 

22 

3400 

5700 

1  27 

25 

1  59 

1   3 

2   6 

25 

9300 

5800 

1  19 

28 

1  48 

0  57 

2  10 

'28 

9200 

5900 

1  11 

31 

1  38 

0  51 

2  15 

32 

9100 

6000 

1   3 

34 

1  28 

0  46 

2  19 

35 

9000 

6100 

0  56 

37 

1  19 

0  40 

2  23 

38 

8900 

6200 

0  49 

39 

I  10 

0  35 

2  27 

40 

8806 

6300 

0  33 

42 

1   1 

0  30 

2  31 

43 

8700 

6400 

I  36 

44 

0  53 

0  26 

2  35 

46 

8600 

6500 

0  31 

47 

0  46 

0  21 

2  38 

48 

8500 

6600 

0  26 

49 

0  39 

0  18 

2  41 

50 

8400 

6700 

0  21 

51 

0  33 

0  14 

2  43 

52 

8300 

6800 

0  17 

52 

0  28 

0  11 

2  46 

54 

8200 

6900 

0  13 

54 

0  23 

0   8 

2  48 

55 

8100 

7000 

0  10 

55 

0  19 

0   6 

2  50 

56 

8000 

7100 

0   7 

56 

0  16 

0   4 

2  51 

57 

7900 

720C 

0   5 

57 

0  13 

0   2 

2  52 

58 

7800 

7300 

0   4 

57 

0  11 

0   1 

2  53 

59 

7700 

740?) 

0   3 

58 

0  10 

0   1 

2  54 

59 

7600 

7500 

0   3 

1  58 

0  18 

0   1 

2  54 

59 

7500 

TABLE    XXIII. 

EQUATIONS  8  TO  9  OF  MOON's  LONGITUDE. ARGUMENTS  8  TO  9. 


Arg. 

3 

9 

Arg. 

8 

9 

/; 

»   // 

»   it 

/   // 

0 

20 

1  20 

5000 

20 

1  SO 

100 

15 

1  29 

5100 

24 

1  26 

200 

11 

1   37 

5200 

29 

1  31 

300 

7 

1  46 

5300 

33 

37 

400 

2 

1  54 

5400 

37 

42 

500 

0  58 

2   1 

5500 

42 

47 

600 

0  54 

2   8 

5600 

46 

51 

700 

0  50 

2  15 

5700 

50 

1   55 

sao 

0  46 

2  20 

5SOO 

54 

58 

901) 

0  42 

2  25 

5900 

1  58 

2   0 

1000 

0  38 

2  29 

6000 

2   1 

2   1 

1?00 

0  35 

2  32 

6100 

2   5 

2   2 

'  1^00 

0  31 

2  34 

6200 

2   8 

2   2 

1300 

0  28 

2  35 

6300 

2  11 

2   1 

1400 

0  25 

2  35 

6400 

2  14 

59 

1500 

0  33 

2  34 

6500 

2  17 

56 

1600 

0  20 

2  32 

6600 

2  19 

52 

1700 

0  18 

2  29 

6700 

2  22 

48 

1800 

0  16 

2  26 

6800 

2  24 

43 

1900 

0  14 

2  21 

6900 

2  25 

38 

2000 

0  13 

2   ,6 

7000 

2  27 

32 

2100 

0  11 

2  11 

7100 

2  28 

25 

2230 

0  10 

2   4 

7200 

2  29 

18 

2300 

0  10 

58 

7300 

2  30 

11 

2400 

0   9 

51 

7400 

2  30 

4 

2500 

0   9 

43 

7500 

2  31 

0  56 

2600 

0  10 

36 

7600 

2  30 

0  49 

2700 

0  10 

29 

7700 

2  30 

0  42 

2800 

0  11 

22 

7800 

2  29 

0  36 

2900 

0  12 

15 

7900 

2  28 

0  29 

3000 

0  13' 

8 

8000 

2  27 

0  24 

3100 

0  15 

2 

8100 

2  26 

0  18 

3200 

0  16 

0  57 

8200 

2  24 

0  14 

3300 

0  18 

0  52 

8300 

2  22 

0  10 

3400 

0  21 

0  47 

8400 

2  20 

0   8 

3500 

0  23 

0  44 

8500 

2  17 

0   6 

3690 

0  26 

0  41 

8600 

2  15 

0   5 

3700 

0  29 

0  39 

8700 

2  12 

0-  5 

3800 

0  32 

0  38 

8800 

2   9 

0   6 

3900 

0  35 

0  38 

8900 

2   5 

0   8 

4000 

0  39 

0  39 

9000 

2   2 

0  11 

4100 

0  42 

0  40 

9100 

58 

0  15 

4200 

0  46 

0  42 

9200 

54 

0  20 

4300 

0  50 

0  45 

9300 

50 

0  25 

4400 

0  54 

0  49 

9400 

46 

0  32 

4500 

0  58 

0  53 

9500 

42 

0  39 

4600 

1   3 

0  58 

9600 

38 

0  46 

4700 

1   7 

1   3 

9700 

33 

0  54 

4800 

1   11 

1   9 

9800 

29 

1   3 

4900 

1   16 

1   14 

9900 

24 

1   11 

5000 

1   20 

1  20 

10000 

20 

1  20 

EQUATIONS  10  AND  11. 


Arg. 

HI 

11 

Arg. 

10 

- 

0 

10 

10 

500 

10 

3 

10  9  11 

510 

10 

n 

20 

9  12 

520 

9 

n 

30 

• 

13 

530 

9 

12 

40 

7 

14 

540 

8 

13 

50 

7 

15 

550 

8 

14 

60 

6 

Iti 

560 

8 

14 

70 

6117 

570 

8 

15 

80 

5 

17 

580 

7 

15 

90 

5 

18 

590 

7 

15 

100 

5 

18 

600 

7 

16 

110 

4 

IS) 

610 

7 

16 

120 

4 

19 

620 

7 

16 

i:-;o 

4 

IS) 

6ro 

7 

16 

140 

4 

1!) 

640 

7 

15 

150 

4 

IS) 

650 

8 

15 

160 

4 

19 

660 

8  15 

170 

4 

18 

670 

8J14 

180 

5 

18 

680 

9113 

190 

I 

17 

690 

9 

13 

200 

5 

16 

700 

10 

12 

210 

(5 

16 

710 

10 

11 

220 

6 

15 

720 

11 

10 

230 

7 

14 

730 

11 

9 

240 

13 

740 

12 

9 

250 

8 

12 

750 

12 

8 

260 

8 

11 

760 

13 

7 

270 

9 

10 

770 

13 

6 

280 

10 

780 

14 

5 

290 

10 

9 

790 

14 

4 

300 

10 

8 

800 

15 

3 

310 

11 

7 

810 

15 

3 

320 

11 

6 

820 

15 

2 

330 

12 

6 

8"0 

16 

2 

340 

12 

5 

840 

16 

1 

350 

12 

5 

850 

16 

360 

12 

5 

8(50 

16 

370 

13 

4 

870 

16 

380 

13 

4 

880 

16 

390 

13 

4 

890 

16 

400 

13 

4 

BOO  15 

2 

410 

13 

5 

910!  15 

2 

420 

12 

5 

920J15 

3 

430 

12 

5 

9301  14 

3 

440 

12 

6 

940 

14 

4 

450 

12 

6 

950 

13 

5 

460 

11 

7 

960 

13 

6 

470 

11 

8 

970 

12 

7 

480 

11 

8 

980 

11 

8 

490 

10 

9 

990 

11 

9 

500 

10 

10 

1000 

10 

10 

TABLE    XXIII. 

EQUATIONS  12  TO  19. 


Arg.  12  13  14  15|16  17  18  19  Arg 


600  'so 
610!  31 
620  32 
630  33 
640  34 

650  34 
660  35 
670  35 
680  36 
690,36 

700 '37 
710137 


31 

32  28 

33  28 
33 

34  29 


29  17 


1612  14  6!l 

1611)14  Sjl 

1711114  ft  1 

10  15l  5 

18!   •  15  6 


35  3( 

36  30 


5  16 
4  It 
4  1 16 
4  16 


900 


17    810 


800 
790 
780 
770 
760 
750 


32 

QUATION  20 

Arg. 

20 

Arg. 

n 

0 

10 

500 

10 

,11 

510 

20 

12 

820 

30 

13 

530 

40 

13 

540 

50 

14 

550 

60  !  15 

560 

70  1  16 

570 

SO 

16 

580 

90 

17 

590 

100 

17 

600 

110 

17 

610 

120 

17 

620 

130 

17 

C30 

140 

17 

640 

150 

17 

650 

160 

17 

660 

170 

16 

670 

180  1  16 

680 

190 

15 

690 

200 

14 

700 

240 

13 

710 

220 

13 

720 

230 

12 

730 

240 

11 

740 

250 

10 

750 

260 

9 

760 

270 

8 

7VO 

7 

780 

[  290 

6 

79«» 

300 

B 

800 

310 

5 

810 

320 

4 

820 

330 

4 

830 

340 

3 

840 

350 

3 

850 

360 

3 

860 

370 

3 

870 

380 

3 

8SO 

390 

3 

890 

400 

3 

900 

410 

g 

910 

420 

4 

920 

430 

4 

930 

440 

5 

940 

450 

6 

950 

460 

6 

960 

470 

7 

970 

480 

8 

980 

490 

9 

990 

500 

10 

.OOOJ 

36  TABLE    XXIV. 

EVECTION.     Argument — Evection  Corrected. 


Os 

Is 

Us 

Ills 

IVs 

Vs 

GO 

1°30'  0" 

2310'  43" 

2°  40  10" 

20  50'  25" 

2039'  8" 

2°  9'  42" 

1 

31  25 

2  11  57 

2  40  51 

2  50  23 

2  38  25 

2  8  29 

2 

32  51 

2  13  9 

2  41  30 

2  50  20 

2  37  40 

2  7  16 

3 

34  16 

2  14  21 

2  42  8 

2  50  15 

2  36  55 

262 

4 

35  42 

2  15  31 

2  42  45 

2  50  9 

2  36  8 

2  4  47 

5 

37  7 

2  16  41 

2  43  21 

2  50  1 

2  35  19 

2  3  32 

6 

38  32 

2  17  50 

2  43  55 

2  49  52 

2  34  30 

2  2  16 

7 

39  57 

2  18  58 

2  44  27 

2  49  41 

2  33  40 

2  1  0 

8 

41  21 

2  20  5 

2  44  59 

2  49  29 

2  32  48 

1  59  43 

9 

42  46 

2  21  11 

2  45  29 

2  49  15 

2  31  55 

1  58  26 

10 

44  10 

2  22  17 

2  45  57 

2  49  0 

2  31  2 

1  57  8 

11 

45  34 

2  23  21 

2  46  24 

2  48  43 

2  30  7 

1  55  49 

12 

46  58 

2  24  24 

2  46  50 

2  48  26 

2  29  11 

54  30 

13 

48  21 

2  25  26 

2  47  14 

2  48  6 

2  28  14 

53  11 

14 

49  44 

2  26  28 

2  47  37 

2  47  45 

2  27  16 

51  51 

15* 

51  7 

2  27  28 

2  47  59 

2  47  23 

2  26  17 

50  31 

16 

52  29 

2  28  27 

2  48  19 

2  47  n 

2  25  17 

49  11 

17 

53  51 

2  29  25 

2  48  37 

5  4b  35 

2  24  16 

47  50 

18 

55  12 

2  30  21 

2  48  54 

2  46  8 

2  23  14 

46  29 

19 

56  33 

2  31  17 

2  49  10 

2  45  41 

2  22  11 

45  7 

20 

57  53 

2  32  11 

2  49  24 

2  45  12 

2  21  7 

43  46 

21 

59  13 

2  33  5 

2  49  37 

2  44  41 

2  20  2 

42  24 

22 

2  0  32 

2  33  57 

2  49  48 

2  44  9 

2  18  56 

41  2 

23 

2  1  51 

2  34  48 

2  49  58 

2  43  36 

2  17  50 

39  39 

24 

239 

2  35  38 

2  50  6 

2  43  2 

2  16  43 

38  17 

25 

2  4  26 

2  36  26 

2  50  13 

2  42  26 

2  15  34 

36  54 

26 

2  5  43 

2  37  13 

2  50  19 

2  41  49 

2  14  25 

35  32 

27 

2  6  59 

2  37  59 

2  50  23 

2  41  11 

2  13  16 

34  9 

23 

2  8  15 

2  38  44 

2  50  25 

2  40  31 

2  12  5 

32  46 

29 

2  9  30 

2  39  28 

2  50  26 

2  39  50 

2  10  54 

31  23 

30 

2  10  43 

2  40  10 

2  50  25 

2  39  8 

2   9  42 

30  0 

>    TABLE    XXV. 
MOON'S  EQUATORIAL  PARALLAX.     Argument. — Arg.  of  the  Evection. 


Os 

U 

Hs 

Ills 

IVs 

Vs 

0° 

1  28" 

1'  23" 

'  9" 

0'  50" 

0'  32" 

0'  18" 

30° 

2 

1  28 

1  22 

8 

0  49 

0  30 

0  18 

28 

4 

1  28 

22 

7 

0  47 

0  29 

0  17 

26 

6 

1  28 

21 

5 

0  46 

0  J8 

0  17 

24 

8 

1  28 

20 

4 

0  45 

0  «.!7 

0  16 

22 

10 

1  28 

19 

3 

0  44 

0  26 

0  16 

20 

12 

1  27 

18 

2 

0  42 

0  25 

0-  15 

18 

14 

1  27 

17 

0 

0  41 

0  24 

0  15 

16 

16 

1  27 

16 

0  59 

0  40 

0  24 

0  15 

14 

18 

26 

15 

0  58 

0  39 

0  23 

0  14 

12 

20 

26 

14 

0  57 

0  37 

0  22 

0  14 

10 

22 

25 

13 

0  55 

0  36 

0  21 

0  14 

8 

24 

25 

12 

0  54 

0  35 

0  20 

0  14 

6 

26 

24 

11 

0  53 

0  34 

0  20 

0  14 

4 

28 

24 

10 

0  51 

0  33 

0  19 

0  13 

2 

30 

23 

9 

0  50 

0  32 

0  18 

0  13 

0 

XI* 

Xs 

IXs 

VIlIs 

VIIs 

VI* 

TABLE   XXIV. 
EVECTION.     Argument. — Evection  Corrected. 


37 


Vis 

VIIs 

VIIIs 

IXs 

Xs 

XIi 

QO 

1030  0" 

00  50'  18" 

GO  20'  52" 

00  9'  34" 

GO  19'  50" 

Oo  49'  16" 

1 

1  28  37 

0  49  6 

0  20  10 

0  9  34 

0  23  32 

0  50  30 

2 

1  27  14 

0  47  55 

0  19  29 

0  9  35 

0  21  16 

0  51  45 

3 

25  51 

0  46  44 

0  18  49 

0  9  37 

0  22  I 

0  53  1 

4 

24  28 

0  45  34 

0  18  11 

0  9  41 

0  22  47 

0  54  17 

5 

23  6 

0  44  26 

0  17  34 

0  9  47 

0  23  34 

0  55  33 

6 

.21  43 

)  43  17 

0  16  58 

0  9  54 

0  24  22 

0  56  51 

7 

20  20 

0  42  10 

0  16  24 

0  10  2 

0  25  12 

0  58  9 

8 

1H  58 

0  41  4 

0  15  50 

0  10  12 

0  26  3 

0  59  28 

9 

17  36 

0  39  58 

0  15  19 

0  10  23 

0  26  55 

1   0  47 

10 

16  14 

0  33  53 

0  14  48 

0  10  36 

0  27  43 

2  7 

11 

14  52 

0  37  49 

0  14  19 

0  10  50 

0  28  43 

3  27 

12 

13  31 

0  36  46 

0  13  51 

0  11  5 

0  29  39 

4  48 

13 

12  10 

0  35  44 

0  13  25 

0  11  23 

0  30  35 

6  9 

14 

10  49 

0  34  43 

0  13  0 

0  11  41 

0  31  33 

7  31 

15 

9  29 

0  33  43 

0  12  37 

0  12  1 

0  32  32 

8  53 

16 

8  09 

0  32  44 

0  12  14 

0  12  23 

0  33  32 

10  16 

17 

6  49 

0  31  46 

0  11  54 

0  12  45 

0  34  34 

11  39 

18 

5  30 

0  30  49 

0  11  34 

0  13  10 

0  35  36 

13  2 

19 

4  11 

0  29  53 

0  11  16 

0  13  35 

0  36  39 

14  26 

20 

2  52 

0  28  58 

0  11  0 

0  14  3 

0  37  43 

15  50 

21 

1  34 

0  28  5 

0  10  45 

0  14  31 

0  38  48 

17  14 

22 

1   0  17 

0  27  12 

0  10  31 

0  15  1 

0  39  55 

18  39 

23 

0  59  0 

0  26  20 

0  10  19 

0  15  33 

0  41  2 

20  3 

24 

0  57  44 

0  25  30 

0  10  8 

0  16  5 

0  42  10 

21  28 

25 

0  56  23 

0  24  40 

0  9  59 

0  16  39 

0  43  19 

22  53 

26 

0  55  13 

0  23  52 

0  9  51 

0  17  15 

0  44  29 

24  18 

27 

0  53  58 

0  23  5 

0  9  45 

0  17  52 

0  45  39 

25  44 

28 

0  52  44 

0  22  20 

0  9  40 

0  18  30 

0  46  51 

27  9 

29 

0  51  31 

0  21  35 

0  9  36 

0  19  9 

0  48  3 

28  34 

30 

0  50  18 

0  20  52 

0  9  34 

0  19  50 

0  49  16 

30  0 

TABLE    P. 
MOON'S  EQUATORIAL  PARALLAX.     Argument. — Arg.  of  tbe  Variation. 


0. 

I. 

Us 

Ills 

IVs 

Vs 

O1 

56" 

42" 

16" 

4" 

18" 

44" 

300 

2 

55 

41 

14 

4 

19 

46 

28 

4 

55 

39 

13 

4 

21 

47 

26 

6 

55 

37 

12 

4 

23 

48 

24 

8 

55 

35 

10 

5 

24 

50 

22 

10 

54 

34 

9 

6 

26 

51 

20 

12 

53 

32 

8 

6 

28 

52 

18 

14 

52 

30 

7 

7 

30 

53 

16 

16 

51 

28 

6 

8 

32 

54 

14 

18 

50 

26 

6 

9 

34 

55 

12 

20 

49 

24 

5 

10 

35 

55 

10 

22 

48 

23 

4 

12 

37 

56 

8 

24 

47 

21 

4 

13 

39 

56 

6 

26 

45 

19 

4 

14 

41 

57 

4 

28 

44 

18 

4 

16 

42 

57 

2 

30 

42 

16 

4 

18 

44 

57 

0 

XIs 

x« 

1X8 

VIIIs 

VIIi 

VI,  | 

38  TABLE    XXV. 

EQUATION  OF  MOON'S  CENTER.      Argument. — Anomaly  corrected. 


Os 

Is 

III 

Ills 

IVa 

Vs 

00 

70  o'  0" 

10°  20'  58" 

120  38'  44" 

13°  17'  35" 

120  16'  21" 

9058  29" 

1 

775 

10  26  52 

12  41  43 

13  17  5 

12  12  48 

9  52  58 

2 

7  14  10 

10  32  42 

12  44  35 

13  16  28 

12  9  11 

9  47  24 

3 

7  21  15 

10  38  27 

12  47  20 

13  15  44 

12  5  29 

9  41  48 

4 

7  28  19 

10  44  8 

12  49  59 

13  14  53 

12  1  41 

9  36  10 

5 

7  35  23 

10  49  43 

12  52  30 

13  13  56 

11  57  49 

9  30  29 

6 

7  42  26 

10  55  14 

12  54  55 

13  12  52 

11  53  52 

9  24  46 

7 

7  49  28 

11   0  39 

12  57  12 

13  11  41 

11  49  50 

9  19  1 

8 

7  56  28 

11   6  0 

12  59  23 

13  10  24 

11  45  44 

9  13  13 

9 

8  3  28 

11  11  15 

13  1  26 

13  9  1 

11  41  33 

9  7  24 

10 

8  10  26 

11  16  24 

13  3  23 

13  7  31 

11  37  17 

9   1  32 

11 

8  17  22 

11  21  29 

13  5  12 

13  5  54 

11  32  57 

8  55  39 

12 

8  24  17 

11  26  27 

13  6  55 

13  4  12 

11  28  33 

8  49  44 

13 

8  31  10 

11  31  20 

13  8  30 

13  2  23 

11  24  5 

8  43  47 

14 

8  38  1 

11  36  8 

13  9  59 

13  0  27 

11  19  32 

8  37  49 

15 

8  44  50 

11  40  49 

13  11  20 

12  58  26 

11  14  55 

8  31  49 

16 

8  51  36 

11  45  25 

13  12  34 

12  56  18 

11  10  14 

8  25  48 

17 

8  58  20 

11  49  54 

13  13  41 

12  54  5 

11   5  30 

8  19  46 

18 

9  5  1 

11  54  18 

13  14  41 

12  51  45 

11   0  41 

8  13  42 

19 

9  11  39 

11  58  35 

13  15  34 

12  49  19 

10  55  49 

8  7  38 

20 

9  18  15 

12  2  47 

13  16  20 

12  46  47 

10  50  53 

8   1  32 

21 

9  24  47 

12  6  52 

13  16  59 

12  44  10 

10  45  53 

7  55  26 

22 

9  31  16 

12  10  50 

13  17  31 

12  41  27 

10  40  50 

7  49  18 

23 

9  37  42 

12  14  42 

13  17  56 

12  38  38 

10  35  43 

7  43  10 

24 

9  44  4 

12  18  23 

13  18  14 

12  35  43 

10  30  33 

7  37  1 

25 

9  50  23 

12  22  7 

13  18  24 

12  32  43 

10  25  20 

7  30  52 

26 

9  56  38 

12  25  40 

13  18  28 

12  29  37 

10  20  4 

7  24  42 

27 

10  2  49 

12  29  6 

13  18  25 

12  26  26 

10  14  45 

7  18  32 

23 

10  8  56 

12  32  25 

13  18  16 

12  23  10 

10  9  22 

7  12  21 

29 

10  14  59 

12  35  38 

13  17  59 

12  19  48 

10  3  57 

7   6  11 

30 

10  20  58 

12  38  44 

13  17  35 

12  16  21 

9  58  29 

700 

TABLE    XXVI. 

MOON'S  EQUATORIAL  PARALLAX.      Argument. — Corrected  Anomaly. 


Os 

Is 

Us 

Ills 

IVi 

Vs 

i 

Oo 

58'  58" 

58'  27" 

57'  8  " 

55'  30" 

54'  2" 

53'  3" 

300 

2 

53  58 

58  23 

57   2 

55  23 

53  57 

53   0 

28 

4 

58  57 

58  19 

56  55 

55  17 

53  52 

52  58 

26 

6 

58  56 

58  14 

56  49 

55  11 

53  47 

52  56 

24 

8 

58  55 

58  10 

56  42 

55   4 

53  43 

52  54 

22 

10 

58  54 

58   5 

56  36 

54  58 

53  38 

52  52 

20 

,  12 

58  53 

58   0 

56  29 

54  52 

53  34 

52  50 

18 

14 

58  51 

57  55 

56  22 

54  46 

53  30 

52  49 

16 

16 

58  49 

57  49 

56  16 

54  40 

53  26 

52  47 

14 

18 

53  46 

57  44 

56   9 

54  34 

53  22 

52  46 

12 

20 

58  44 

57  38 

56   3 

54  29 

53  19 

52  45 

10 

22 

58  41 

57  32 

55  56 

54  23 

53  15 

52  44 

S 

24 

58  38 

57  26 

55  49 

54  18 

53  12 

52  43 

6 

26 

58  34 

57  20 

55  43 

54  12 

53   9 

52  43 

4 

28 

58  31 

57  14 

55  36 

54   7 

53   6 

52  43 

2 

30 

58  27 

57   8 

55  30 

54   2 

53   3 

52  43 

0 

XIs 

X* 

IXs 

vm» 

VIIi 

Vis 

TABLE   XXV  39 

EQUATION  OP  MOON'S  CENTER.      Argument. — Anomaly  corrected. 


Vis 

VIIs 

VIIIs 

IXs 

• 

Xs 

Xls 

GO 

70    0'    0" 

40    i'3i» 

10  43'  39  ' 

00  42'  25" 

1021'  16" 

3039'    2" 

1 

G    53  4(J 

3    56    3 

40  12 

0   42     1 

24  22 

3   45     1 

2 

G   47  39 

3   59  38 

36  50 

0   41  44 

27  35 

3    51     4 

3 

6   41  28 

3   45  15 

33  34 

0    41  35 

30  54 

3   57  11 

4 

6    35  18 

3    39  56 

30  23 

0   41  32 

34  20 

4     3  22 

5 

6   21)     8 

3   34  40 

27  17 

0   41  36 

37  53 

4     9  37 

6 

6   22  59 

3   29  26 

24  17 

0   41  46 

41  32 

4    15  55 

4 

G    16  5'J 

3   24  17 

21  22 

0   42    4 

45  18 

4    22  18 

8 

G    10  42 

3    19  10 

18  33 

0   42  29 

49  10 

4   28  44 

9 

G     4  34 

3    14     7 

15  50 

0   43     1 

53     8 

4    35  13 

1) 

5    58  23 

397 

13  12 

0   43  40 

57  13 

4   41  45 

11 

5    52  22 

3     4  11 

10  41 

0   44  26 

2     1  24 

4   43  21 

12 

5   4G  17 

2   59  19 

8  15 

0   45  19 

2     5  42 

4   54  59 

13 

5   4:)  14 

2   54  30 

5  55 

0   46  19 

2    10     5 

5      1  40 

14 

5   34  12 

2   49  46 

3  42 

0   47  26 

2    14  35 

5     8  24 

15 

5    2*  11 

2   45     5 

1  34 

0   48  40 

2    19  11 

5    15  10 

16 

5   22  11 

2   49  28 

0    59  33 

0    50     1 

2   23  52 

5   21  59    , 

17 

5    16  13 

2   35  55 

0   57  37 

0   51  30 

2   24  39 

5   23  50 

18 

5    10  16 

2    31  27 

0   55  48 

0    53    5 

2    33  32 

5    35  43 

19 

5     4  21 

2   27     3 

0    54     6 

0   54  47 

2   38  31 

5   42  37 

2) 

4    58  23 

2   22  43 

0   52  29 

0    56  37 

2    43  35 

5   49  34 

21 

4   52  36 

2    18  27 

0   50  5!) 

0   58  33 

2   48  45 

5    56  32 

22 

4    46  47 

2    14  16 

0    49  36 

1      0  37 

2   54     0 

6     3  31 

2i 

4    4  )  59 

2    10  10 

0   48  19 

1      2  48 

2   59  21 

6    10  32 

21 

4    36  14 

268 

0   47     8 

1      5    5 

3     4  46 

6    17  34 

25 

4   29  31 

2     2  11 

0    46    4 

7  30 

3    10  17 

6   24  37 

26 

4   23  50 

1    58  19 

0    45     7 

10     I 

3    15  52 

6    31  41 

27 

4    18  11 

1    54  31 

0    44  16 

12  40 

3   21  33 

6   38  45 

Si 

4    12  35 

1    50  49 

0    43  32 

15  25 

3   27  18 

6   45  50 

29 

472 

1    47  11 

0    42  55 

18  17 

3    33     8 

6   52  55 

3!) 

4      1   31 

1    43  39 

0   42  -25 

21   16 

3    39    2 

700 

TABLE    XXVI.     (Continued.) 
REDUCTION  OF  PARALLAX,  AND  ALSO  OF  THE  LATITUDE.    Argument. — Latitude. 


Latitude. 

Uedu'Mion  of 
Parallax. 

Reduction  of 
LiitiUii'e. 

3° 

0' 

1'    12' 

(J 

0 

2    2- 

9 

0 

3     :i2 

12 

0 

4     39 

15 

1 

5    4i 

18 

1 

6    44 

21 

1 

7     40 

24 

2 

8     31 

27 

2 

9     16 

30 

3 

9    55 

33 

3 

10    28 

36 

4 

10    54 

39 

5 

11     13 

4-2 

5 

11     25 

45 

6 

11     29 

Latitude. 

Reduction  of 
Parallax. 

Reduction  of 
Latitude. 

48° 

G 

11'     15' 

51 

7 

11     14 

51 

8 

10     56 

57 

8 

10    30 

GO 

9 

9    57 

G3 

9 

9     18 

G6 

10 

8    33 

69 

10 

7     42 

72 

10 

6    46 

75 

11 

5    45 

78 

11 

4    41 

81 

11 

3    33 

84 

11 

2    24 

87 

11 

1     12 

90 

11 

0      0 

40 


TABLE    XXVII. 


VARIATION. 
ARGUMENT. — Variation,  corrected. 


Us 

Is 

Us 

Ills 

IVs 

V» 

o 

o  '   " 

O   '    " 

0   •   a 

O   t   n 

O   '   " 

O   1    H 

0 

0  38  0 

1   8  1 

6  58 

0  35  54 

0  5  29 

062 

2 

0  40  26 

1   9  7 

5  36 

0  33  27 

0  4  21 

0  7  24 

4 

0  42  52 

1  10  3 

4  5 

0  31   0 

0  3  22 

0  8  55 

6 

0  45  16 

1  10  50 

2  27 

0  28  34 

0  2  33 

0  10  34 

8 

0  47  38 

1  11  26 

0  42 

0  26  11 

0  1  54 

0  12  22 

10 

0  49  57 

1  11  53 

0  58  49 

0  23  51 

0  1  24 

0  14  17 

12 

0  52  13 

1  12  9 

0  56  50 

0  21  34 

0  1   5 

0  16  19 

14 

0  54  24 

1  12  15 

0  54  45 

0  19  22 

0  0  57 

0  18  27 

16 

0  56  30 

1  12  10 

0  52  35 

0  17  15 

0  0  59 

0  20  41 

18 

0  58  30 

1  11  55 

0  50  21 

0  15  13 

0  1  11 

0  23  0 

20 

0  24 

I  11  30 

0  48  2 

0  13  17 

0  1  34 

0  25  23 

22 

2  11 

I  10  55 

0  45  40 

0  11  28 

028 

0  27  50 

24 

3  51 

1  10  10 

0  43  16 

0  9  47 

0  2  51 

0  30  20 

26 

5  23 

1   9  15 

0  40  50 

0  8  13 

0  3  45 

0  32  52 

28 

6  47 

1   8  11 

0  38  22 

0   6  47 

0  4  48 

0  35  26 

30 

8  1 

1   6  58 

0  35  54 

0  5  26 

062 

0  38  0 

Vis 

VIIs 

VIIIs 

IXs 

Xs 

XIs 

o 

0       '        " 

0       '         n 

O       '        " 

O       '         " 

O       '        " 

O       '        n 

0 

0   38     0 

9    58 

10   30 

0    40      6 

092 

0     7    58 

2 

0   40   34 

11    11 

9    13 

0    37    38 

0    7     49 

0     9    13 

4 

0   43     8 

12    15 

7    47 

0    35    10 

0    6    45 

0    10   37 

6 

0   45   40 

13      9 

6    13 

0    32   44 

0    5    50 

0    12     9 

8 

0   48    10 

13    52 

4    31 

0    30    19 

055 

0    13    49 

10 

0   50    37 

14   26 

2   42 

0    27    58 

0    4    29 

0    15    36 

12 

0   53     0 

14    48 

0   47 

0    25    39 

044 

0    17    30 

14 

0    55    19 

15      1 

0    58   45 

0    23    25 

0    3    50 

0    19    30 

16 

0   57    33 

15     3 

0    56   38 

0    21    15 

0    3    45 

0   21    36 

18 

0    58    41 

14    54 

0   54    25 

0    19    10 

0    3    51 

0   23   47 

20 

1      1    43 

14    35 

0   52     9 

0    17    11 

047 

0   26     3 

22 

1      3   38 

14      6 

0   49    49 

0    15    18 

0    4    34 

0   28   22 

24 

1      5    25 

13    27 

0   47    26 

0    13    33 

0    5     10 

0    30    44 

26 

1      7      5 

12    38 

0   45     0 

0    11    54 

0    5    57 

0   33     8 

28 

1      8   36 

11    39 

0   42   33 

0    10    24 

0    6    53 

0   35    33 

30 

1      9    58 

10    30 

0   40     6 

092 

0    7    58 

0   38     0 

TABLE   XXVIII. 

MOON'S    DISTANCE    FROM    THE    NORTH    POLE   OF    THE   ECLIPTIC. 
ARGUMENT.     Supplement  of  Node-f-Moon's  Orbit  Longitude. 


41 


Ills 

IVs 

Vs 

vis   i   viis 

VIIIs 

<p 

«4°  ftP7  16" 

85°  20'  43" 

87°  13'  47" 

89°  48'  0"  i  92°  22'  13" 

94°  15'  17" 

300 

a 

84  39  27 

85  26  16 

87  23  12 

89  58  46 

92  31  27 

94  20  31 

28 

4 

84  40   1 

85  32  9 

87  32  48 

90   9  31 

92  40  30 

94  25  25 

26 

84  40  58 

85  38  20 

87  42  33 

90  20  14 

92  49  19 

94  29  59 

24 

8 

84  42  17 

85  44  50 

88  52  28 

90  30  55 

92  57  56 

94  34  12 

22 

10 

84  43  58 

85  51  37 

88   2  31 

90  41  33 

93   6  18 

94  38  4 

20 

12 

84  46  2 

85  58  42 

88  12  42 

90  52  7 

93  14  27 

94  41  35 

18 

14 

84  48  27 

86   6  3 

88  23  0 

91   2  36 

93  22  20 

94  44  45 

16 

16 

84  51  15 

86  13  40 

88  33  24 

91  13  0 

93  29  57 

94  47  32 

14 

18 

84  54  25 

86  21  33 

88  43  53 

91  23  18   93  37  18 

94  49  58 

12 

20 

84  57  56 

86  29  42 

88  54  27 

91  33  29 

93  44  23 

94  52  2 

10 

22 

85   1  48 

86  38  4 

89   5  5 

91  43  32 

93  51  10 

94  53  43 

8 

24 

88   6   1 

86  46  41 

89  15  46 

91  53  27 

93  57  40 

94  55  2 

6 

85  10  35 

86  55  30 

89  26  29 

92  3  12 

94  3  51 

94  55  59 

4 

28 

85  15  29 

87   4  32 

89  37  14 

92  12  48 

94  9  44 

94  56  33 

a 

30 

85  20  48 

87  13  47 

89  48  0 

92  22  13 

94  15  17 

94  56  44 

0 

IIS 

" 

Os 

Xls 

Xs 

IXs 

TABLE    XXIX. 

EQUATION    II    OF    THE   MOON?S   POLAR    DISTANCE. 
ARGUMENT  II,  corrected. 


Ills 

IVs 

V« 

Vis 

VIIs 

VIIIs 

0° 

0'  14" 

1'  24" 

4'  37" 

9'  0" 

13'  23" 

16'  36" 

30° 

2 

0  14 

1  34 

4  53 

9  18 

13  39 

16  45 

28 

4 

0  15 

1  44 

5  9 

9  37 

13  54 

16  53 

26 

6 

0  17 

1  54 

5  26 

9  55 

14  9 

17   1 

24 

8 

0  19 

2  5 

5  43 

10  13 

14  24 

'"*  G 

22 

ie 

0  22 

2  17 

6  0 

10  31 

14  38 

17  14 

20 

12 

0  25 

2  29 

6  17 

10  49 

14  52 

17  20 

18 

14 

0  29 

2  41 

6  35 

11  7 

15  5 

17  26 

16 

16 

0  34 

2  54 

6  53 

11  25 

15  Id 

17  31 

14 

18 

0  40 

3  8 

7  11 

11  43 

15  31 

17  35 

12 

20 

0  45 

3  22 

7  29 

12  0 

15  43 

17  38 

10 

22 

0  52 

3  36 

7  47 

12  17 

15  55 

17  41 

8 

24 

0  59 

3  51 

8  5 

12  34 

16  6 

17  43 

6 

26 

1  7 

4  6 

8  23 

12  51 

16  16 

17  45 

4 

28 

1  15 

4  21 

8  42 

13  7 

16  26 

17  46 

a 

30 

1  24 

4  37 

9  0 

13  23 

16  36 

17  46 

0 

|    ITs 

Is 

Os 

Xls 

Xs 

IXs 

TABLE    XXX. 

EQUATION    III    OF   THE    POLAR    DISTANCE. 
ARGUMENT.     Moon's  True  Longitude. 


Ills 

IVs 

Vs 

Vis 

VIIs 

VIIIs 

0° 

16" 

16" 

12" 

8" 

n 

1" 

30° 

6 

16 

14 

11 

7 

1 

24 

12 

16 

14 

10 

6 

0 

18 

18 

16 

13 

10 

5 

0 

12 

24 

15 

13 

9 

5 

0 

6 

30 

15 

12 

8 

4 

0 

0 

Us 

Is 

Os 

Xls 

Xs 

IXs 

TABLE    XXXI. 


EQUATIONS    OF    POLAR    DISTANCE. 
ARGUMENTS. — 20  of  Longitude;  V  to  IX,  corrected;  and  X,  not  corrected 


Arg. 

20   i   V.     VI. 

VII. 

VIII. 

IX. 

x 

Arg. 

260 

0"  !  56"    6" 

3" 

25" 

3" 

11" 

240 

280 

1 

55 

6 

3 

25 

3 

11 

220 

300 

1 

55 

7 

4 

25 

4 

11 

200 

320 

2 

53 

8 

5 

24 

6 

32 

180 

340 

3 

52 

10 

6 

23 

7 

13 

160 

360 

4 

50 

12 

8 

23 

9 

14 

140 

380 

5 

48 

14 

10 

22 

11 

16 

120 

400 

6 

45 

16 

12 

21 

14 

17 

100 

420 

8 

42 

18 

14 

20 

17 

19 

80 

440 

10 

39 

21 

17 

19 

») 

21 

60 

460 

11 

36 

24 

19 

17 

23 

23 

40 

480 

13 

33 

27 

22 

16 

27 

25 

20 

500 

15 

30 

30 

25 

15 

30 

27 

000 

520 

17 

27 

33 

28 

14 

33 

29 

980 

54!) 

19 

24 

36 

31 

12 

37 

31 

960 

560 

20 

20 

39 

33 

11 

40 

33 

940 

580 

22 

17 

41 

36 

10 

43 

35 

920 

6!)0 

24 

15 

44 

38 

9 

46 

37 

900 

620 

25 

12 

46 

40 

8 

48 

38 

880 

640 

26 

10 

48 

42 

7 

51 

40 

860 

660 

27 

8 

50 

44 

6 

53 

41 

840 

680 

28 

7 

52 

45 

6 

54 

42 

820 

700 

2!) 

5 

53 

46 

5 

56 

42 

800 

720 

21) 

5 

53 

47 

5 

56 

43 

780 

740 

30 

4 

54 

47 

5 

57 

43 

760 

TABLE     XXXII. 

REDUCTION 
ARGUMENT. — Supplement  of  Node  -f-  Moon's  Orbit  Longitude. 


1  Os  Vis 
«' 

I*  VIls 

Us  Vllls 

Ills  IXs 

IVs  Xs 

Vs  XIs 

00  I  7'  0" 

r  3" 

r  3" 

7'  0" 

13'  57' 

12'  57" 

2 

6  31 

0  49 

1  18 

7  29 

13  10 

12  42 

4 

6   3 

0  38 

1  35 

7  57 

13  22 

12  25 

6 

5  34 

0  28 

1  54 

8  26 

13  32 

12   6 

8 

5   6 

0  20 

2  14 

8  54 

13  40 

11  46 

10 

4  39 

0  14 

2  35 

9  21 

13  46 

11  25 

12 

4  12 

0  10 

2  58 

9  48 

13  50 

11   2 

14 

3  46 

0   8 

3  22 

10  13 

13  52 

10  38 

16 

3  22 

0   8 

3  46 

10  38 

13  52 

10  13 

18 

2  58 

0  10 

4  12 

11   2 

13  50 

9  48 

20 

2  35 

0  14 

4  39 

11  25 

13  46 

9  21 

22 

2  14 

0  20 

5   6 

11  46 

13  40 

8  54 

24 

1  54 

0  28 

5  34 

12   6 

13  32 

8  26 

26 

1  35 

0  38 

6   3 

12  25 

13  22 

7  57 

28 

1  18 

0  49 

6  31 

12  42 

13  10 

7  29 

30  |  1   3 

1   3 

7   0 

12  57 

12  57 

7   0 

TABLE  XXXIV. 

MOON'S   SEMIDIAMETER. 
ARGUMENT.     Equatorial  Parallax. 


43 


Eq.  Parallax. 

Semidiam. 

Eq.  Parallax. 

Semidiam. 

Eq.  Parallax. 

Semidiam 

53'  0" 

14'  27" 

56'  0" 

15'  16" 

W  0" 

16'  5" 

53  20 

14  32 

56  20 

15  21 

59  20 

16  10 

53  40 

14  37 

56  40 

15  26 

59  40 

16  16 

54  0 

14  43 

:  57  0 

15  32 

60  0 

16  21 

64  20 

14  48  . 

57  20 

15  37 

60  20 

16  26 

54  40 

14  54 

57  40 

15  43 

60  40 

16  32 

55  0 

14  59 

58  0 

15  48 

61  0 

16  37 

55  20 

15  5 

58  20 

15  54 

61  20 

16  43 

55  40 

15  10 

58  40 

15  59 

61  40 

16  48 

56  0 

15  16 

59  0 

16  5 

63  0 

16  54 

TABLE    XXXV. 


TABLE    XXXVI. 


AUGMENTATION   OF   MOON*S   SEMI-          MOON*S   HOURLY   MOTION   IN   LON- 


DIAMETER. 
ARGUMENT.     Apparent  Altitude. 


Ap.  Alt. 

Augm. 

6° 

2" 

12 

3 

18 

5 

24 

6 

30 

8 

36 

9 

42 

11 

48 

12 

54 

13 

60 

14 

66 

15 

72 

15 

.     78 

16 

84 

16 

90 

16 

GITUDE. 

ARGUMENTS.    2,  3,  4,  and  5  of  Lon- 
gitude. 


Arg. 

2 

3 

4 

5 

Arg. 

0 

6" 

u 

3' 

3 

100 

5 

5 

3 

3 

95 

10 

5 

3 

3 

90 

15 

4 

3 

3 

85 

20 

4 

2 

2 

80 

25 

3 

a 

2 

75 

30 

2 

2 

a 

70 

35 

2 

1 

1 

65 

40 

1 

1 

1 

60 

45 

1 

1 

1 

55 

50 

0 

1 

1 

60 

TABLE    XXXVII. 

MOON'S     HOURLY    MOTION     IN     LONGITUDE 
ARGUMENT.     Argument  of  the  Erection. 


Os 

Is 

III 

Ills 

IVs 

Vs 

0» 

1'  20" 

'  15" 

i  o" 

0'  39" 

20' 

0'  6" 

30° 

2 

1  20 

14 

58 

0  38 

19 

0 

28 

4 

1  20 

13 

57 

0  37 

18 

0 

26 

6 

1  20 

12 

56 

0  35 

16 

0 

24 

8 

1  20 

11 

54 

0  34 

15 

0 

22 

10 

1  20 

11 

53 

0  33 

14 

0 

20 

12 

1  19 

10 

M 

0  31 

13 

0 

18 

14 

1  19 

9 

50 

0  30 

ia 

0 

16 

16 

1  19 

8 

49 

0  29 

11 

0 

14 

18 

1  18 

7 

48 

0  27 

11 

0 

ia 

20 

1  18 

5 

46 

0  26 

10 

0 

10 

2? 

1  17 

4 

45 

0  25 

9 

8 

24 

1  17 

3 

44 

0  23 

0 

6 

26 

1  16 

2 

42 

0  23 

0 

4 

28 

1  15 

1 

41 

0  21 

0 

a 

30 

1  15 

0 

39 

0  20 

0 

0 

XI. 

Xt 

IXs 

VIIIs 

vm 

Vis 

26 


44  TABLE  XXXVIII. 

MOON'S   HOURLY   MOTION   IN   LONGITUDE. 
ARGUMENTS.    Sum  of  preceding  equations,  and  Anomaly,  correct-  d. 


0" 

20" 

40" 

60" 

80" 

100" 

Os    0° 

n 

6" 

9" 

11" 

14" 

16" 

XIIs   OP 

10 

11 

13 

M 

n 

20 

11 

13 

15 

ifl 

Is     0 

11 

13 

15 

XIs    0 

10 

11 

13 

14 

20 

20 

11 

12 

13 

10 

iis   o 

8 

11 

12 

13 

Xs    0 

10 

9 

10 

10 

11 

12 

20 

20 

10 

10 

10 

10 

11 

10 

THs   0 

10 

10 

10 

10 

10 

10 

IXk    0 

10 

11 

11 

10 

10 

20 

20 

12 

11 

10 

10 

10 

IVs   0 

13 

12 

11 

9 

vin  o 

10 

14 

12 

11 

9 

20 

20 

14 

12 

11 

9 

10 

*Vs    0 

15 

13 

11 

9 

Vlli   0 

10 

15 

13 

11 

9 

5 

20 

X 

15 

13 

11 

9 

5 

10 

Vis   0 

15 

13 

11 

9 

5 

Vis    0 

0" 

20" 

40" 

60" 

80" 

100" 

TABLE    XXXIX. 

MOON'S     HOURLY    MOTION     IN    LONGITUDE. 
ARGUMENT.     Anomaly,  corrected. 


Os 

Is 

Us 

Ills 

IVs 

Vs 

jfi 

34'  51" 

34'  14" 

32'  39" 

30'  45" 

29*  6" 

28'  1" 

3flP 

i 

34  51 

34  9 

32  32 

30  38 

29  0 

27  58 

28 

4 

34  51 

34  4 

32  24 

30  31 

28  55 

27  55 

26 

34  50 

33  59 

32  17 

30  23 

28  60 

27  53 

24 

8 

34  49 

33  53 

32  9 

30  16 

28  45 

27  50 

22 

10 

34  47 

33  47 

32  2 

30  9 

28  40 

27  48 

20 

12 

34  45 

33  41 

31  54 

30  2 

28  35 

27  46 

18 

14 

34  43 

33  35 

31  46 

29  56 

28  30 

27  45 

16 

16 

34  41 

33  28 

31  38 

29  49 

28  26 

27  43 

14 

18 

34  38 

33  22 

31  31 

29  42 

28  22 

27  42 

12 

20 

34  34 

33  15 

31  23 

29  36 

28  18 

27  41 

10 

22 

34  31 

33  8 

31  15 

29  30 

28  14 

27  40 

8 

84 

34  2? 

33  1 

31   8 

29  23 

28  10 

27  39 

6 

26 

34  23 

32  54 

31  0 

29  17 

28  7 

27  39 

4 

28 

34  19 

32  47 

30  53 

29  12 

28  4 

27  38 

2 

30 

34  14 

32  39 

30  45 

29  6 

28  1 

27  38 

0 

XIs 

x» 

IXs 

VHIs 

VIls 

Vis 

1ABLE  XL.  45 

MOON'S    HOURLY    MOTION     IN    LONGITUDE. 
ARGUMENTS.    Sum  of  preceding  equations,  and  Argument  of  Variation. 


27' 

29' 

81' 

33' 

35' 

87' 

Os          0° 

0" 

2" 

6" 

7" 

1C" 

12" 

xn«     o» 

10 

0 

12 

90 

20 

1 

11 

10 

Is           0 

3 

9 

XIs         0 

10 

5 

20 

20 

7 

10 

III         0 

9 

XB          0 

10 

11 

20 

20 

12 

1 

10 

m«     o 

12 

1 

IXs         0 

10 

12 

1 

20 

20 

11 

10 

IVs        0 

9 

vim    o 

10 

5 

20 

20 

7 

10 

Vs         0 

9 

VIIi       0 

10 

11 

20 

20 

10 

12 

10 

Vis        0 

2 

10 

12 

Vis        0 

27' 

29' 

31' 

33' 

35' 

37' 

TABLE    XLI. 

MOON'S    HOURLY    MOTION    IN    LONGITUDE. 
ARGUMENT.     Argument  of  the  Variation. 


Os 

I« 

Hs 

Ills 

IVs 

Vs 

0° 

1'  17" 

0'  58" 

V  20" 

0*  2" 

0'  22" 

i  0" 

»• 

2 

1  17 

0  55 

18 

3 

24 

2 

88 

4 

17 

0  53 

16 

3 

26 

4 

26 

6 

16 

0  51 

14 

3 

29 

6 

24 

8 

16 

0  48 

12 

4 

31 

8 

22 

10 

15 

0  45 

11 

6 

34 

10 

20 

12 

14 

0  43 

9 

6 

37 

12 

18 

14 

13 

0  40 

8 

7 

39 

13 

16 

16 

11 

0  3R 

8 

8 

42 

15 

14 

18 

10 

0  35 

5 

10 

44 

16 

12 

20 

8 

0  32 

4 

11 

47 

17 

10 

22 

6 

0  30 

4 

13 

50 

18 

8 

24 

4 

0  27 

3 

15 

52 

18 

6 

26 

2 

0  25 

3 

17 

55 

19 

4 

28 

0 

0  23 

2 

19 

57 

19 

i 

30 

0  58 

0  20 

2 

22 

0 

19 

0 

Xts 

Xs 

IXs 

VIIIs 

VIIs 

Vis 

i6 


TABLE  XLII. 

MOON'S     HOURLY    MOTION     IN    LONGITUDE. 
Argument  of  the  Reduction. 


0> 

Is 

Us 

in* 

IVi 

Vi 

OP 

2" 

6" 

14" 

18" 

14" 

6" 

SOP 

2 

2 

7 

14 

18 

13 

6 

28 

4 

2 

7 

15 

18 

13 

5 

26 

6 

2 

8 

15 

18 

12 

5 

24 

8 

2 

8 

16 

18 

13 

4 

22 

10 

2 

9 

16 

17 

11 

4 

20 

12 

3 

9 

16 

17 

11 

4 

18 

14 

3 

10 

17 

17 

10 

3 

16 

16 

3 

10 

17 

17 

10 

3 

14 

18 

4 

11 

17 

16 

3 

12 

20 

4 

11 

17 

16 

3 

10 

22 

4 

12 

18 

16 

2 

8 

34 

5 

12 

18 

15 

2 

6 

as 

5 

13 

18 

15 

2 

4 

28 

6 

13 

18 

14 

2 

2 

30 

6 

14 

18 

14 

2 

0 

XIs 

Xs 

IXs 

VIIIs 

VIIi 

VI§ 

TABLE    XLIII. 

MOON'S    HOURLY    MOTION     IN    LATITUDE. 
ARGUMENT.     Argument  I,  of  Latitude. 


08+ 

b+ 

Ils-f- 

Ills— 

IVs— 

Vs— 

0° 

2*  58" 

*  34" 

'  29" 

(X  0" 

1'  29" 

y  34" 

39> 

2 

2  58 

2  31 

24 

6 

1  35 

2  37 

• 

4 

2  58 

2  23 

18 

12 

1  40 

2  40 

26 

6 

2  57 

2  24 

13 

19 

1  45 

2  43 

24 

8 

2  56 

2  20 

7 

25 

1  50 

2  45 

23 

10 

2  55 

2  17 

31 

1  55 

2  47 

20 

12 

2  54 

2  12 

0  55 

37 

1  59 

2  49 

18 

14 

2  53 

2  8 

0  49 

43 

2  4 

2  51 

16 

16 

2  51 

2  4  - 

0  43 

49 

2  8 

2  53 

14 

18 

2  49 

59 

0  37 

55 

2  12 

2  54 

13 

20 

2  47 

55 

31 

1 

2  17 

2  55 

10 

22 

2  45 

50 

25 

7 

2  20 

2  56 

9 

24 

2  43 

45 

19 

13 

2  24 

2  57 

6 

26 

2  40 

40 

12 

18 

2  28 

2  58 

4 

28 

2  37 

35 

6 

24 

2  31 

2  58 

3 

30 

2  34 

1  29 

0 

29 

2  34 

2  58 

t 

XIs+ 

x»+ 

1X8+ 

VIIIs— 

VIIs— 

Vis— 

TABLE    XLIV. 

MOON'S    HOURLY     MOTION     IN     LATITUDE. 
ARGUMENT.    Argument  II,  of  Latitude. 


0.+ 

I.+ 

II.+ 

IIIs— 

ITi— 

Vt— 

(P 
« 

12 
18 
84 

30 

4" 

4" 

2" 

0" 
0 

1 

2 
2 

2" 
3 
3 
8 
3 
4 

4" 

300 

24 
18 
13 
< 

t 

XI.+ 

X.+ 

IXt-f 

VIII*— 

VII»— 

VI.— 

TABLE    XLV.— PROPORTIONAL  LOGARITHMS.      47 


0 

1' 

2' 

3' 

4' 

5' 

—  r 

6' 

V 

0" 

00000 

17782 

14771 

13010 

11761 

10792 

10000 

9331 

1 

35663 

17710 

14735 

12986 

11743 

10777 

9988 

9320 

2 

82553 

17639 

14699 

12962 

11725 

10763 

9976 

9310 

3 

80792 

17570 

14664 

12939 

11707 

10749 

9964 

9300 

4 

29542 

17501 

14629 

12915 

11689 

10734 

9952 

9289 

5 

28573 

17434 

14594 

12891 

11671 

10720 

9940 

9279 

6 

87782 

17368 

14559 

12868 

11654 

10706 

9928 

9269 

7 

27112 

17302 

14525 

12845 

11636 

10692 

9916 

9259 

8 

26532 

17238 

14491 

12821 

11619 

10678 

9905 

9249 

9 

26021 

17175 

14457 

12798 

11601 

10663 

9893 

9238 

10 

25563 

17112 

14424 

12775 

11584 

10649 

9881 

9228 

11 

25149 

17050 

14390 

12753 

11666 

10635 

9S69 

9218 

12 

24771 

16960 

14357 

12730 

11549 

10621 

9858 

9208   i 

13 

24424 

16930 

14325 

12707 

11532 

10608 

9846 

9198 

14 

24102 

16871 

14292 

12685 

11515 

10594 

9834 

9188 

15 

23802 

16812 

14260 

12663 

11498 

10580 

9826 

9178 

16 

23522 

16755 

14228 

12640 

11481 

10566 

9811 

9168 

17 

23259 

16698 

14196 

12«18 

11464 

10552 

9800 

9158 

18 

23010 

16642 

14165 

12596 

11457 

10539 

9788 

9148 

19 

22775 

16587 

14133 

12574 

11430 

10525 

9777 

9138 

20 

22553 

16532 

14102 

12553 

11413 

10512 

9765 

9128 

21 

22341 

16478 

14071 

12531 

11397 

10498 

9754 

9119 

22 

22139 

16425 

14040 

12510 

11380 

10484 

9742 

9109 

23 

21946 

16372 

14010 

12488 

11363 

10471 

9731 

9099 

24 

21761 

16320 

13979 

12467 

11347 

10458 

9720 

9089 

25 

21584 

162ti9 

13949 

12445 

11331 

10444 

9708 

9079 

26 

21413 

16218 

13919 

12424 

11314 

10431 

9697 

9070 

27 

21249 

13168 

13890 

12403 

11298 

10418 

9686 

9060 

28 

21091 

16118 

13860 

12382 

11282 

10404 

9tf75 

9050 

29 

20939 

16069 

13831 

12362 

11266 

10391 

9664 

9041 

30 

20792 

16021 

13802 

12341 

11249 

10378 

9652 

9031 

31 

20649 

15973 

13773 

12320 

11233 

10365 

9641 

9021 

32 

20512 

15925 

13745 

12300 

11217 

10352 

9630 

9012 

33 

20378 

15878 

13716 

12279 

11201 

10339 

9619 

9002 

34 

20248 

15832 

13688 

12259 

11186 

10326 

9608 

QQQQ 

35 

20122 

15786 

13660 

12239 

11170 

10313 

9597 

8983 

36 

20000 

15740 

13632 

12218 

11154 

10300 

9586 

8973  J 

37 

19881 

15695 

13604 

12198 

11138 

10287 

9575 

8964 

38 

19765 

15651 

13576 

12178 

11123 

10274 

9564 

8954 

39 

19652 

15607 

13549 

12159 

11107 

10261 

9553 

8945 

40 

19542 

15563 

13522 

12139 

11091 

10248 

9542 

8935 

41 

19435 

15520 

13495 

12119 

11076 

10235 

9532 

8926 

42 

19331 

15477 

13468 

12099 

11061 

10223 

9521 

8917 

43 

19228 

15435 

13441 

12080 

11045 

10210 

9510 

8907 

44 

19128 

15393 

13415 

12061 

11030 

10197 

9499 

8898 

45 

19031 

15351 

13388 

12041 

11015 

10185 

9488 

8888 

46 

18935 

15310 

13362 

12022 

10999 

10172 

9476 

8879 

47 

18842 

15269 

13336 

12003 

10984 

10160 

9467 

8870' 

48 

18751 

15229 

.  13310 

11984 

10969 

10147 

9456 

8861 

49 

18661 

15189 

13284 

11965 

10954 

10135 

9446 

8851 

50 

18573 

15149 

13259 

11946 

10939 

10122 

9435 

8842 

51 

18487 

15110 

13233 

11927 

10924 

10110 

9425 

8833 

62 

18403 

15071 

13208 

11908 

10909 

10098 

9414 

8824 

53 

18320 

15032 

13183 

11889 

10894 

10085 

9404 

8814 

54 

18239 

149*1 

13158 

11871 

10880 

10073 

9393 

&305 

55 

1«T>9 

14956 

13133 

11852 

10865 

10061 

9383 

8796 

56 

18081 

14918 

13108 

11834 

10850 

10049 

9372 

8787 

57 

18004 

14881 

13083 

11816 

10835 

10036 

9362 

8778      i 

58 

17929 

14844 

13059 

11797 

10821 

10024 

9351 

8769      - 

59 

17855 

14808 

13034 

11779 

10806 

10012 

9341 

8760 

W 

17782 

14771 

13010 

11761 

10792 

10000 

9331 

8751      | 

48      TABLE   XLV.— PROPORTIONAL  LOGARITHMS. 


* 

8' 

9' 

10' 

11' 

12' 

13' 

14' 

15' 

»  j 

0" 

8751 

8239 

7782 

7368 

6990 

6642 

6320 

6021 

5740 

1 

8742 

8231 

7774 

7361 

6984 

6637 

6315 

6016 

5736 

2 

8733 

8223 

7767 

7354 

6978 

6631 

6310 

6011 

5731 

8724 

8215 

7760 

7348 

6972 

6625 

6305 

6006 

6727 

8715 

8207 

7753 

7341 

6966 

6620 

6300 

6001 

5722 

8706 

8199 

7745 

7335 

6960 

6614 

6294 

5997 

6718 

8697 

8191 

7738 

7328 

6954 

6609 

6289 

6992 

6713 

8688 

8183 

7731 

7322 

6948 

6803 

6284 

6987 

6709 

8679 

8175 

7724 

7315 

6942 

6598 

6279 

5982 

5704 

8670 

8167 

7717 

7309 

6936 

6592 

6274 

1   5977 

6700 

10 

8661 

8159 

7710 

7302 

6930 

6587 

6269 

6973 

6695 

11 

8652 

8152 

7703 

7296 

6924 

6581 

6364 

6968 

6691 

12 

8643 

8144 

7696 

7289 

6918 

6576 

6259 

6963 

6686 

13 

8635 

8136 

7688 

7283 

6912 

6570 

6254 

6958 

6682 

14 

8626 

8128 

7681 

7276 

($06 

6565 

6243 

6954 

5677 

15 

8617 

8120 

7674 

7270 

6900 

6559 

6243 

5949 

6673 

16 

febus 

8112 

7667 

7264 

6894 

6554 

6238 

5944 

6669 

17 

855? 

8104 

7660 

7257 

6888 

6548 

6233 

5939 

6664 

18 

85H1 

8097 

7653 

7251 

6882 

6543 

6228 

59;J5 

6660 

19' 

8582 

8089 

7646 

7244 

6877 

6538 

6223 

6930 

56S5 

20 

8573 

8081 

7639 

7238 

6871 

6532 

6218 

6925 

6651 

21 

8565 

8073 

7632 

7232 

6865 

6527 

6213 

5920 

5646 

22 

8556 

8066 

7625 

7225 

6859 

621 

6208 

5916 

5642 

23 

8547 

8058 

7618 

7219 

685:} 

6516 

6203 

5911 

5637 

24 

8539 

8050 

7611 

7212 

6847 

6510 

6193 

5fH)6 

6633 

25 

8530 

8043 

7604 

7206 

6841 

63)3 

6193 

6902 

5629 

26 

8522 

8035 

7597 

7200 

6836 

6500 

6188 

5897 

5624 

27 

8513 

8027 

7590 

7193 

6830 

6494 

6183 

5892 

5H20 

28 

8504 

8020 

7583 

7187 

682-1 

6489 

6178 

5888 

£615 

29 

8496 

8012 

7577 

7181 

6818 

64S4 

6173 

5883 

5611 

30 

8487 

8004 

7570 

7175 

6812 

6478 

6168 

6878 

5607 

31 

8479 

7997 

7563 

7168 

6807 

6473 

6163 

5874 

5602 

32 

8470 

7989 

7556 

7162 

6801 

6467 

6158 

5ft>9 

5598 

33 

8462 

7981 

7549 

7156 

6795 

6462 

6153 

5864 

5c94 

34 

8453 

7974 

7542 

7149 

6789 

6457 

6148 

58HO 

5589 

;   35 

8445 

7966 

7535 

7143 

6784 

6451 

6143 

5855 

5585 

|  36 

8437 

7959 

7528 

7137 

6778 

6-1-16 

6138 

5850 

5580 

37 

8428 

7951 

7522 

7131 

6772 

6141 

6133 

5846 

5576 

I   38 

8420 

7944 

7515 

7124 

6766 

6435 

6128 

5841 

5572 

00 
39 

8411 

7936 

7508 

7118 

6761 

6430 

6123 

58136 

5567 

40 

8403 

7929 

7501 

7112 

6755 

6425 

6118 

5832 

5563 

« 

8395 

7921 

7494 

7106 

6749 

6420 

6113 

5827 

5559 

42 

8386 

7914 

7488 

7100 

6743 

6414 

610S 

5823 

5554 

43 

8378 

7906 

7481 

7093 

6738 

64U9 

6103 

5S18 

5550 

44 

8370 

7899 

7474 

7087 

6732 

6404 

601)9 

5813 

5546 

45 

8361 

7891 

7467 

7081 

6726 

6398 

6094 

5S09 

5541 

46 

8353 

7884 

7461 

7075 

6721 

6393 

60S9 

5F04 

5537 

47 

8345 

7877 

7454 

7069 

6715 

638S 

60  H4 

5800 

6533 

48 

8337 

7869 

7447 

7063 

6709 

6333 

6079 

5795 

6528 

49 

8328 

7862 

7441 

7057 

6704 

6377 

6074 

5790 

5524 

50 

8320 

7855 

7434 

7050 

6372 

6069 

6786 

6520 

51 

8312  , 

7847 

7427 

7044 

6692 

6067 

6064 

6781 

6516  - 

52 

8304 

7840 

7421 

703S 

6687 

6362 

6u:,9 

6777 

5511 

53 

8296 

7832 

7414 

7032 

6681 

6357 

6(155 

5772 

6507 

54 

8288 

7825 

7407 

7036 

6676 

6351 

6050 

6768 

6503 

55 

8279 

7818 

7401 

70-JO 

6670 

6346 

6045 

6763 

6498 

56 

8271 

7811 

7394 

7014 

6664 

6341 

6040 

6758 

6494 

67 

8263 

7803 

7387 

7008 

6659 

633b 

60:35 

6754 

5490 

68 

8255 

7796 

7381 

7002 

6653 

63'<1 

6(180 

5749 

5488 

59 

8247 

7789 

7374 

6996 

6648 

6325 

6025 

6745 

6481 

M 

8239 

7782 

7368 

6990 

6642 

6320 

6021 

5740 

6477 

TABLR   XLV.— PROPORTIONAL   LOGARITHMS        49 


17' 

18 

19' 

20' 

21' 

22' 

23' 

24' 

25' 

9 

5477 

5229 

4994 

4771 

4559 

4357 

4164 

3979 

3802 

1 

5473 

5225 

4990 

4768 

4556 

4354 

4161 

3976 

379S» 

2 

5«9 

5221 

4986 

4764 

4552 

4351 

4158 

3973 

3796 

5-164 

5217 

4983 

4760 

4549 

4347 

4155 

3970 

3793 

5460 

5213 

4979 

4757 

4546 

4344 

4152 

3967 

3791 

6456 

5209 

4975 

4753 

4543 

4341 

4149 

3964 

3788 

5452 

52v)5 

4971 

4750 

4539 

4338 

4145 

3961 

3785 

5447 

5201 

4967 

4746 

4535 

4334 

4142 

3958 

3782 

5443 

5197 

4964 

4742 

4532 

4331 

4139 

3955 

3779 

5439 

5193 

4960 

4739 

4528 

4328 

4l:« 

S952 

3776 

10 

5435 

5189 

4956 

4735 

4525 

4325 

4133 

3949 

3773 

11 

5430 

5185 

4952 

4732 

4522 

4321 

4130 

3946 

3770 

12 

5426 

5181 

4949 

4728 

4518 

4318 

4127 

3943 

3768 

13 

54:32 

5177 

4945 

4724 

4515 

4315 

4124 

3940 

3765 

14 

5418 

5173 

4941 

4721 

4511 

4311 

4120 

3937 

3762 

15 

5414 

5169 

4937 

4717 

4508 

4308 

4117 

3934 

3759 

16 

5409 

5165 

4933 

4714 

4505 

4305 

4114 

3931 

3756 

17 

5405 

51fl 

4930 

4710 

4501 

4302 

4111 

3928 

3753 

18 

5401 

5157 

4926 

4707 

4498 

4298 

4108 

3925 

3750 

19 

5397 

5153 

4922 

4703 

4494 

4295 

4105 

3922 

3747 

20 

5393 

5149 

4918 

4699 

4491 

42»2 

4102 

8919 

3745 

21 

5389 

5145 

4915 

4696 

4488 

4289 

4099 

3917 

3742 

22 

5384 

5111 

4911 

4692 

4484 

4285 

40% 

3914 

3739 

23 

5380 

5137 

4907 

4689 

4481 

4282 

4092 

3911 

8736 

24 

5376 

5133 

4903 

4685 

4477 

4279 

4089 

3908 

3733 

25 

5372 

5129 

4900 

4682 

4474 

4276 

4086 

3905 

3730 

26 

5368 

5125 

4896 

4678 

4471 

4273 

4083 

3902 

3727 

27 

5364 

5122 

48H2 

4675 

4467 

4269 

40SO 

3899 

3725 

28 

5359 

5118 

4889 

4671 

4464 

4266 

4077 

3a96 

3722 

29 

5355 

5114 

4885 

4668 

4460 

4263 

4074 

3893 

3719 

30 

5351 

5110 

4881 

4664 

4457 

4260 

4071 

3890 

3716 

31 

5347 

5106 

4877 

4660 

4454 

4256 

4068 

3887 

3713 

32 

5343 

5102 

4874 

4657 

4450 

4253 

4065 

3884 

3710 

33 

5339 

5098 

4870 

4653 

4447 

4250 

4062 

3881 

3708 

34 

5335 

5C94 

4866 

4650 

4444 

4247 

4059 

3878 

3705 

35 

5331 

5090 

4863 

4646 

4440 

4244 

4055 

3875 

3703 

36 

5326 

5086 

4859 

4643 

4437 

4240 

4052 

3872 

3699 

37 

5322 

5032 

4855 

4639 

4434 

4237 

4049 

3869 

3696 

38 

5318 

5079 

4852 

4636 

4430 

4234 

4046 

3866 

3693 

39 

5314 

5075 

4848 

4632 

4427 

4231 

4043 

3863 

3691 

40 

531U 

5071 

4844 

4629 

4424 

4228 

4040 

3860 

3688 

41 

5306 

5067 

4841 

4625 

4420 

4224 

4037 

3857 

3685 

42 

5302 

5063 

48)7 

4622 

4417 

4221 

4034 

3855 

3682 

43 

5298 

5059 

4833 

4618 

4414 

4218 

4031 

8852 

3679 

44 

5294 

5055 

4830 

4615 

4410 

4215 

4028 

3849 

3677 

45 

5290 

5051 

4826 

4611 

4407 

4212 

4025 

3846 

3674 

46 

5285 

5048 

4822 

4608 

4404 

4209 

4022 

3843 

3671 

47 

5281 

5044 

4819 

4604 

4400 

4205 

4019 

3S40 

3668 

48 

5277 

5040 

4815 

4601 

4397 

4202 

4016 

3837 

3665 

49 

5273 

5036 

4811 

4597 

4304 

4199 

4043 

3834 

3663 

50 

5269 

5032 

4808 

4594 

4390 

4196 

4010 

3831 

3660 

51 

5265 

5028 

4804 

4590 

4387 

4193 

4007 

3828 

3657 

52 

6261 

5025 

4800 

4587 

4384 

4189 

4004 

3825 

3654 

53 

5257 

5021 

4797 

4584 

4380 

4186 

4001 

3922 

3651 

54 

5253 

5017 

4793 

4580 

4377 

4183 

3998 

3820 

3649 

55 

5249 

5013 

4789 

4577 

4374 

4180 

3995 

3317 

3616 

H 

5245 

o009 

4786 

4573 

4370 

4177 

3991 

3814 

3643 

57 

52-11 

5005 

4782 

4570 

4367 

4174 

3988 

3811 

3640 

58 

5237 

5002 

4778 

4566 

4364 

4171 

3985 

8808 

3637 

59 

5233 

4998 

4775 

4563 

4361 

4167 

3982 

3805 

3635 

60 

5229 

4994 

4771 

4559 

4357 

4164 

3979 

4802 

3633 

50      TABLE  XLV.— PROPORTIONAL  LOGARITHMS. 


27' 


29' 


30' 


31' 


32' 


33' 


34' 


3623 
3621 
3618 

3615 
3612 
3610 
3607 
3604 

3601 


3590 
3587 


3576 


3574 
3571 


3555 


3544 
3541 


3535 


3533 
3530 


3525 


3519 
3516 
3514 
3511 


3503 
3500 
3497 


3484 


3479 
3476 
3473 
3471 
3468 


3465 
3463 
3460 
3457 
3454 

3452 
3449 
3446 
3444 
3441 

3438 
3436 
3433 
3431 


3423 
3420 
3-117 
3415 

3412 
3409 
3407 
3404 
3401 


3307 
3305 


3300 


3274 
3271 


3264 


3248 


3241 


3378 


3372 
3370 
3367 


3357 


3346 
3344 
£341 


3313 


3218 


3213 
3210 


3203 
3200 
3198 
3195 


3190 


3180 
3178 
3175 
3173 
3170 


3160 


3158 
3155 
3153 
3150 
3148 
3145 

3143 
3140 
3138 
3135 


3130 


3123 
3120 

3118 
3115 
3113 
3110 
3108 

3105 
3103 
3101 

3098 
3096 


3091 


3078 
3076 
3073 
3071 


3064 
3061 


3056 
3054 
3052 
3049 
3047 

3044 
3042 
3039 
3037 


3010 
3008 
3005 
3003 
3001 


2991 


2977 
2974 


2972 


2955 
2953 
2950 


2943 


2934 


2915 


2847 


2730 
2728 
2725 
2723 
2721 
2719 

2716 
2714 
2712 
2710 
2707 

2705 
2703 
2701 


3018 
3015 
3013 
3010 


2877 


2817 
2815 


2S10 


2801 


2796 
2794 


2787 

2785 
2782 
2780 
2778 
2775 

2773 
2771 
2769 
2766 


2762 
2760 
2757 
2755 
2753 

2750 
2748 
2746 
2744 
2741 


2737 


2732 
2730 


2678 


2672 
2669 
2667 


2594 


2581 
2579 
2577 
2574 

2572 
2570 


2564 
2561 


2544 
2542 

2540 


2467 
2465 


2454 
2452 
2450 
2448 
2445 

3443 
2441 
2439 
2437 


2431 
2429 
2426 
2424 


2418 
2416 
2414 

2412 
2410 
2408 
2405 


2401 


2647 
2645 
2643 
2640 


2634 


2616 
2614 
2812 
2610 
2607 


2518 
2516 
2514 
2512 
2510 


2501 


2490 


2477 

£475 
2473 
2471 
2469 


2374 
2372 


2370 


2351 


2347 


234? 
2341 


TABLE   XLV—  PROPORTIONAL   LOGARITHMS.     51 


35' 

2341 
2339 
2?37 
2335 
2233 
2331 


2324 
2322 
2320 

2318 
2316 
2314 
2312 
2310 


2294 


2279 

2277 
2275 
2273 
2271 


2261 
2259 


2257 


2231 


36' 

2218 
2216 
2214 
2212 
2210 
2J08 

220(5 
2204 
2202 
2200 
2198 

2196 
2194 
2192 
2190 

2188 


2184 
2182 
2180 
2178 

2176 
2174 
2172 
2170 


2167 
2165 
2163 
2161 


2157 
2155 
2153 
2151 
2149 

2147 
2145 
2143 
2141 


2137 
2135 


2131 


2127 
2125 
2123 
2121 
2119 

2117 
2115 
2113 
2111 
2109 

2107 
2J05 
2103 

2IUJ 


37' 


2086 
2084 
2082 


2078 
2076 
2074 
2072 
2070 


2066 


2062 
2061 


2057 
2055 
9053 
2051 

2049 
2047 
2045 
2043 
2041 


2037 


2024 


2018 
2016 
2014 
2012 

2010 
2039 
2007 
2005 


2001 
1999 

1997 


1984 


38' 


1982 
1980 
1978 
1976 
1974 

1972 
1970 
1968 
1967 
1965 

1963 
1961 
1959 
1957 
1955 


1951 
1950 
1948 
1946 

1944 
1942 
1940 
1988 
1936 

1934 
1933 
1931 


1919 
1918 

1916 
1914 
1912 
1910 
1908 

1906 
1904 
1903 
1901 


1897 
1895 


1891 


1884 


1878 
1876 
1875 
1873 
1871 


39' 

1871 

1869 

1867 


1854 


1850 
1849 
1847 
1845 


1841 

1839 
1838 
1836 
1834 

1832 
1830 


1821 
1819 
1817 
1816 

1814 
1812 
1810 

1808 
1806 

1805 
1803 
1801 
1799 
1797 

1795 
1794 
1792 
1790 
1788 

1786 
1785 
1783 
1781 
1779 

1777 
1775 
1774 
1772 
1770 

1768 
1766 
1765 
1763 
1761 


1761 
1759 
1757 
1755 
1754 
1752 

1750 
1748 
1746 
1745 
1743 

1741 

1739 
1737 
1736 
1734 

1732 
1730 
1728 
1727 
1725 

1723 
1721 
1719 

1718 
1716 

1714 
1712 
1711 

1709 
1707 

1705 
1703 
1702 
1700 


1687 
1686 

1684 


1678 
1677 
1675 
1673 
1671 

1670 


1663 

1661 
1(559 
1657 
1655 
1654 


41' 

1654 
1652 
1650 
1648 
1647 
1645 

1643 
1641 
1640 
1638 
1636 

1634 
1633 
1631 
1629 
1627 


1617 
1615 
1613 
1612 
1610 

1608 
1606 
1605 
1603 
1601 


1596 


1589 

1587 
1585 
1584 


1578 
1577 
1575 

1573 
1571 
1570 
1568 
1566 

1565 
1563 
1561 
1559 
1558 


1554 
1552 
155* 
1549 


42' 

1549 
1547 
1546 
1544 
1542 
1540 

1539 
1537 
1535 
1534 
1532 

1530 
1528 
1527 


1518 
1516 
1515 

1513 
1511 
1510 

1508 
1506 

1504 
1503 
1501 
1499 
1498 

1496 
1494 
1493 
1491 


1487 


1481 

1479 
1477 
1476 
1474 
1472 

1470 
1469 
1467 
1465 
1464 


1460 
1459 
1457 
1455 

1454 
1452 
1450 
1449 
1447 


43' 

1447 
1445 
1443 
1442 
1440 
1438 

1437 
1435 
1433 
1432 
1430 

1428 
1427 


1420 
1418 
1417 
1416 
1413 

1412 
1410 

1408 
1407 
1405 

T403 
1402 
1400 
1398 
1397 

1395 


1390 
1388 


1387 


1382 
1380 

1378 
1377 
1375 
1373 
1372 

1370 


1363 

1362 
1360 
1359 
1357 
1355 

1354 
1352 
1350 
1349 
1347 


52       TABLE    XLV.— PROPORTIONAL   LOGARITHMS. 


44'     45- 

46' 

47' 

48' 

49' 

50' 

51' 

52* 

0" 

1347     1249 

1154 

1061 

P69 

880 

792 

706 

621 

1 

1345   !   1248 

1152 

1059 

968 

878 

790 

704 

620 

2 

1344     1246 

1151 

1057 

966 

877 

789 

703 

619 

3 

1342     1245 

1149 

1056 

965 

875 

787 

702 

617 

4 

1340     1^3 

1148 

1054 

963 

874 

786 

700 

616 

5 

1339     1241 

1146 

1053 

962 

872 

785 

699 

615 

6 

1337     1240 

1145 

1051 

960 

871 

783 

697 

613 

7 

1335     1238 

1143 

1050 

959 

769 

782 

696 

612 

8 

1334 

1237 

1141 

1048 

957 

868 

780 

694 

610 

9 

1332 

1235 

1140 

1047 

956 

866 

779 

693 

609 

10 

1331 

1233 

1138 

1045 

954 

865 

777 

692 

608 

11 

1329 

1232 

1137 

1044 

953 

863 

776 

690 

606 

12 

1327 

1230 

1135 

1042 

951 

862 

774 

689 

605 

13 

1326 

1229 

1134 

1041 

950 

860 

773 

687 

603 

14 

1324 

1227 

1132 

1039 

948 

859 

772 

686 

602 

15 

1322 

1225 

1130 

1037 

947 

857 

770 

685 

601 

16 

1321 

1224 

1129 

1036 

945 

856 

769 

683 

599 

17 

1319 

1222 

1127 

1034 

944 

855 

767 

682 

598 

1 

1317 

1221 

1126 

1033 

942 

853 

766 

680 

596 

19 

1316 

1219 

1124 

1031 

941 

852 

764 

679 

595 

20 

1314 

1217 

1123 

1030 

939 

850 

763 

678 

594 

21 

1313 

1216 

1121 

1028 

938 

849 

762 

676 

592 

22 

1311 

1214 

1119 

1027 

936 

847 

760 

675 

591 

23 

1309 

1213 

1118 

1025 

935 

846 

759 

673 

590 

24 

1308 

1211 

1116 

1024 

933 

844 

757 

672 

588 

25 

1306 

1209 

1115 

1022 

932 

843 

756 

670 

587 

26 

1304 

1208 

1113 

1021 

930 

841 

754 

669 

685 

27 

1303 

1206 

1112 

1019 

929 

840 

753 

668 

584 

28 

1301 

1205 

1110 

1018 

927 

838 

751 

666 

583 

29 

1300 

Ii03 

1109 

1016 

926 

837 

750 

665 

581 

30 

1298 

1201 

1107 

1015 

924 

835 

749 

663 

680 

31 

1296 

1200 

1105 

1013 

923 

834 

747 

662 

579 

32 

1295 

1198 

1104 

1012 

921 

833 

746 

661 

577 

33 

1293 

1197 

1102 

1010 

920 

831 

744 

659 

576 

34 

1291 

1195 

1101 

1008 

918 

830 

743 

658 

574 

35 

1290 

1193 

1099 

1007 

917 

828 

741 

656 

673 

36 

1288 

1192 

k]098 

1005 

915 

827 

740 

655 

572 

37 

1287 

1190 

1096 

1004 

914 

825 

739 

654 

570 

38 

1285 

1189 

1095 

1002 

912 

824 

737 

652 

569 

39 

1283 

1187 

1093 

1001 

911 

822 

736 

651 

568 

40 

1282 

1186 

1091 

999 

909 

821 

734 

649 

666 

41 

1280 

1184 

1090 

998 

908 

819 

733 

648 

665 

42 

1278 

1182 

1088 

996 

906 

818 

731 

647 

563 

43 

1277 

1181 

1087 

995 

905 

816 

730 

645 

562 

44 

1275 

1179 

1085 

993 

903 

815 

729 

644 

561 

45 

1274 

1178 

1084 

992 

902 

814 

727 

642 

559 

46 

1272 

1176 

1082 

990 

900 

812 

726 

641 

558 

47 

1270 

1174 

1081 

989 

899 

811 

724 

640 

657 

48 

1269 

1173 

1079 

987 

897 

809 

723 

638 

655 

49 

1267 

1171 

1078 

986 

896 

808 

721 

637 

664 

50 

1266 

1170 

1076 

984 

894 

806 

720 

635 

652 

51 

1264 

1168 

1074 

983 

893 

805 

719 

634 

651 

62 

1262 

1167 

1073 

981 

891 

803 

717 

633 

650 

63 

1261 

1165 

1071 

980 

890 

802 

716 

631 

648 

64 

1259 

1163 

1070 

978 

888 

801 

714 

680 

647 

65 

1257 

1162 

1068 

977 

877 

799 

713 

628 

646 

86 

1256 

1160 

1067 

975 

885 

798 

711 

627 

644 

57 

1254 

1159 

1065 

974 

884 

796 

710 

626 

553 

68 

1253 

1157 

1064 

972 

883 

795 

709 

624 

641 

m 

1251 

fcll$6 

1063 

971 

881 

793 

707 

623 

640 

60 

1249 

1154 

1061 

9ti8 

880 

792 

706 

621 

639 

TABLE    XLV.— PROPORTIONAL   LOGARITHMS*      03 


53' 

54' 

55'   i   56' 

57' 

58' 

59' 

0" 

539 

458 

378 

300 

220 

147 

73 

1 

537 

456 

377 

298 

221 

146 

72 

2 

536 

455 

375 

297 

2JO 

145 

71 

3 

5:35 

454 

374 

296 

219 

143 

69 

4 

533 

452 

373 

294 

218 

142 

68 

5 

532 

451 

371 

2J3 

216 

141 

67 

6 

531 

450 

370 

292 

815 

140 

66 

7 

529 

448 

369 

291 

U14 

139 

64 

8 

523 

447 

367 

289 

!)13 

137 

63 

9 

526 

446 

366 

288 

5111 

136 

62 

10 

525 

444 

365 

287 

felO 

135 

61 

11 

524 

443 

363 

285 

209 

134 

60 

12 

522 

442 

362 

284 

208 

132 

58 

13 

521 

440 

3H1 

283 

£06 

131 

57 

14 

520 

439 

359 

282 

205 

130 

56 

15 

518 

438 

358 

280 

204 

129 

55 

A 

517 

436 

357 

279 

202 

127 

53 

17 

516 

435 

356 

278 

201 

126 

52 

18 

514 

434 

354 

276 

200 

125 

51 

19 

513 

432 

353 

275 

199 

124 

50 

20 

512 

431 

352 

274 

U7 

122 

49 

21 

510 

430 

350 

273 

t» 

121 

47 

22 

509 

428 

349 

271 

195 

120 

46 

23 

507 

427 

348 

270 

K4 

119 

45 

24 

506 

426 

346 

269 

15-2 

117 

44 

25 

505 

424 

345 

267 

191 

116 

42 

26 

503 

423 

344 

2.56 

19o 

115 

41 

27 

502 

422 

342 

2H5 

189 

114 

40 

28 

501 

420 

341 

2u4 

187 

112 

39 

29 

499 

419 

340 

2v2 

186 

111 

38 

30 

498 

418 

339 

2bl 

185 

110 

36 

31 

497 

416 

337 

280 

184 

109 

35 

32 

495 

415 

336 

258 

182 

107 

34 

33 

494 

414 

335 

257 

181 

106 

33 

34 

493 

412 

333 

256 

180 

105 

31 

35 

491 

411 

332 

255 

179 

104 

30 

36 

490 

410 

331 

253 

177 

103 

29 

37 

489 

408 

329 

252 

176 

101 

23 

38 

487 

407 

328 

251 

175 

100 

27 

39 

486 

406 

327 

250 

174 

99 

25 

40 

484 

404 

326 

248 

172 

98 

24 

41 

483 

403 

324 

247 

171 

96 

23 

42 

482 

402 

323 

246 

170 

95 

22 

43 

4SO 

400 

322 

244 

169 

94 

21 

44 

479 

399 

320 

243 

167 

93 

19 

45 

478 

398 

319 

242 

166 

91 

18 

46 

476 

396 

318 

241 

165 

90 

17 

47 

475 

395 

316 

239 

163 

89 

16 

48 

474 

394 

315 

238 

162 

88 

15 

49 

472 

392 

314 

237 

Ifil 

87 

13 

50 

471 

391 

313 

235 

160 

85 

12 

51 

470 

390 

311 

334 

158 

84 

11 

52 

468 

388 

310 

233 

157 

b3 

10 

53 

467 

387 

309 

232 

156 

82 

8 

54 

466 

3*6 

307 

230 

155 

80 

7 

55 

464 

384 

306 

2-jy 

153 

79 

6 

56 

463 

M 

805 

223 

i58 

78 

5 

57 

462 

3.S2 

304 

227 

151 

77 

4 

58 

4tX) 

381   i   302 

225 

150 

75 

a 

59 

459 

379     301 

224 

148 

74 

1 

60 

458 

378     300 

223 

147 

73 

0 

TABLES. 


SATELLITES  OF  JUPITER. 


Sat. 

Mean  Distance. 

Sidereal  Revolu- 
tion. 

Inclination  of 
Orbit  to  that  of 
Jupiter. 

Mags  ;  that  of 
Jupiter  being 
1000000000. 

1 

2 
3 
4 

6  04853 
9.62347 
15.35024 
26.99835 

d.       h.       m. 
1      18      28 

3     13     14 
7      3    43 
16     16    32 

O        '        " 

3      5    30 

Variable. 
Variable. 
2    58    48 

17328 
23235 
88497 
42659 

SATELLITES  OF  SATURN. 


Sat. 

Mean 
Distance. 

Sidereal  Revolu- 
tion. 

—  i 
Eccentricities  and  Inclinations. 

d.          h.        m. 

The  orbits  of  the  six  inte- 

1 

33.51 

0    22    38 

rior  satellites  are  nearly  cir- 

2 

4.300 

1      8    53 

cular,  and  very  nearly  in  the 

3 

5.284 

1     21     18 

plane   of  the  ring.     That  of 

4 

6.819 

2     17    45 

the  seventh    is   considerably 

5 

9.524 

4     12    25 

inclined  to  the  rest,  and  ap- 

6 

7 

22.081 
64.359 

15    22    41 
79      0    55 

proaches  nearer  to  coincidence 
with  the  ecliptic. 

SATELLITES  OF  URANUS. 


Sat. 

Mean 
Distance. 

Sidereal  Period. 

Inclination  to  Ecliptic. 

Their    orbits   are   inclined 

1? 

13  120 

5    21     25      0 

about  78°  58'  to  the  ecliptic, 

2 

17.022 

8     16    56      5 

and  their  motion  is  retrograde. 

3? 
4 
5? 

19.845 
22.752 
45.507 

10    23      4      0 
13     11       8    59 
38      1    48      0 

The  periods  of  the  2d  and  4th 
require  a  trifling  correction. 
The  orbits  appear  to  be  nearly 

6? 

91.008 

107    16    40      0 

circles. 

TABLE  OF  ASTEROIDS. 


THE      ASTEROIDS. 


The  following  tabular  facts  are  from  the  most  reliable 
sources — the  English  Nautical  Almanac  and  other  Euro- 
pean publications. 


Planets. 

Mean  dis- 
tance from 
the  sun. 

Mean  time 
of  Revolu- 
tion. 

Eccentri- 
city of 
orbits. 

Lon.  of  the 
Ascending 
node. 

Inclination 
of 
orbit. 

Flora    

Earth's  dis.  1. 

22014 

Days. 
1193  16 

0  15677 

deg.     m. 
110     21 

To  Ecliptic. 

5°  53' 

Victoria  

23348 

1303  08 

021854 

2N5    40 

8    23 

*  Vesta,  

2.3627 

132H.26 

0.08945 

103    24 

7     08 

23858 

1345  6b 

0  23232 

259    44 

5    28 

Metis,  

2.3868 

1346.90 

0.12274 

68    28 

5    36 

Hebe  

2.4256 

1379.68 

0  20200 

138    32 

14    47 

Parthenope,  .  .  . 

2.4483 
2.5H05 

1399.06 
1515.40 

0.09800 
0.16974 

125    00 

86    51 

4    37 

9    06 

Astrea   

26173 

1547  58 

0  18880 

141    28 

5     19 

E-o-eria 

25829 

151582 

0  08628 

43    18 

16    33 

2.6679 

1591.68 

0.256.37 

170    58 

13    03 

•Ceres    

27653 

167986 

0  07904 

80    49 

10    36 

*Pallas  

2.7715 

1686.22 

0.23894 

172    37 

34    42 

2  6483 

157408 

0  18856 

293    54 

11     44 

3.1512 

2043.38 

0.10090 

287    38 

3    47 

Psyche 

29771 

1834  61 

0  13082 

150    36 

3    04 

Fortuna 

25342 

1  440  80 

0  17023 

211    17 

1     32 

Melpomene,  
Thetis  

2.3292 

2.4718 

126981 
1419.31 

0.21644 
0.12736 

150    00 
125    25 

10     09 
5     35 

Lutetia 

2  4353 

1387  77 

0  16104 

80    34 

3    05 

'Calliope,  
Amphitrite,.   .  . 

2.9054 
2.5521 

1809.00 
1489.22 

0.10308 
0.06716 

66    38 
356   27 

13    45 

«    08 

*  We  made  an  effort  to  arrange  these  planets  in  the  order  of  their 
distances  from  the  sun,  and  we  have  done  so,  as  far  as  Hygeia  The 
following  ones  were  subsequent  discoveries.  Some  future  day,  when 
their  elements  will  be  better  known,  by  more  varied  and  extended 
observations,  a  re -arrangement  can  be  made. 


• 


s 


•*. 


BOOK  IS  D0E 


T 


ow 


•• 


